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GLOSSARY

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

This Glossary is an alphabetized list of words and phrases used in Mathematical English.  It includes several kinds of entries:

• Technical words in math that have two or more widely-used definitions.
• Technical words that suffer from semantic contamination.
• Words such as if, and, and all that are used to communicate the logical structure of math arguments. These are also discussed on the math reasoning page, but all are cross referenced. 

Most technical words in math are not included.  The best sources for technical words and phrases in math are:

• MathWorld
• PlanetMath
• Wikipedia

A

a, an

The word "a" or "an" is the indefinite article, one of two articles in English.

The articles in English are the indefinite article "a" (with variant "an") and the definite article "the". If your native language is not English you may have problems under­standing mathe­matical dis­course because the articles are used in mathe­matical English in some fairly subtle ways.

Generic use

In math writing, the indefinite article may be used with the name of a type of math object to produce an indefinite des­cription – a des­crip­tion that indicates an arbitrary or unspecified object of that type. (If you use "the", you get a definite descrip­tion that refers to an instance of the object that has been previously referred to.)

When an indefinite description occurs
in the subject of a clause in math English,
it usually must be read as referring to
every object that fits that description.

Example

"Show that an integer that is divisible by four is divisible by two."

• Correct interpretation: Show that every integer that is divisible by four is divisible by two.
• Incorrect interpretation: Show that some integer that is divisible by four is divisible by two.

So in a sentence like this the indefinite article has the force of a universal quanti­fier. More examples are given in the entries on universal quanti­fier and conditional assertions.

More examples

• "A number divisible by $4$ is even." (Subject of sentence.) This means that any number divisible by $4$ is even.
• "Show that a number divisible by $4$ is even." (Subject of subordinate clause.) Again, it means any number.
• "Problem: Find a number divisible by $4$." (Object of verb.) This does not mean find every number divisible by $4$; one will do.

Difference with ordinary English

In ordinary English sentences, such as a "A wolf takes a mate for life." the meaning is that the assertion is true for a typical individual (typical wolf in this case). In math, however, the assertion is required to be true without exception. 

abstract algebra

See algebra.

abuse of notation

Notation may be called abuse of notation if it involves

• suppression of parameters
• synecdoche
• identifying two structures along an isomorphism between them.

Click on those entries for examples.

The word "abuse" makes it sound worse than it really is.  Without judicious use of this technique much mathematical writing would be unreadable.

aleph

algebra

This word has many different meanings in the school system of the USA, and college math majors in particular may be confused by the differences.

High school and college math are changing rapidly,
so these descriptions are not dependable.

• High school algebra is primarily algorithmic and concrete in nature.  This is where you learn to solve linear and quadratic equations (MW, Wik) and to apply them using word problems.
• College algebra is the name given to a college course, perhaps remedial, covering the material covered in high school algebra.
• Linear algebra may be a course in matrix theory (MW, Wik) or a course in linear transformations in a more abstract setting.
• A college course for math majors called algebra, abstract algebra, or perhaps modern algebra, is an introduction to groups, rings, fields and perhaps modules (MW, Wik). It is for many students the first course in abstract mathematics and may play the role of a filter course. In some departments, linear algebra plays the role of the first course in abstraction.
• Universal algebra is a subject math majors don't usually see except possibly in graduate school. It is the general theory of structures with n-ary operations subject to equations, and is quite different in character from abstract algebra.

algorithm

The word algorithm is used in three confusingly similar ways:

algorithm as process

Mathematicians typically use the word "algorithm" for a step by step process for calculating something, as for example the procedure expressed roughly by the description (NM) below.  People who use the word "algorithm" in this way may refer to a program implementing it as the code for the algorithm.

algorithm as program

A program may itself be called "an algorithm".  In my experience, this is not common usage.

algorithm as mathematical object

In computer science or logic texts, the word "algorithm" may be given a mathematical definition (for example as a Turing machine (MW, Wik)), turning an algorithm into a mathematical object.

Example

You can write a program in Python and another one in C to take a list $a:=(a_1,\,a_2,\,a_3,\,a_4\,\ldots)$ with at least three entries and swap the second and third entries. There is a sense in which the two programs, although different as programs, implement the "same" algorithm (process): Change $(a_1,\,a_2,\,a_3,\,a_4\,\ldots)$ to $(a_1,\,a_3,\,a_2,\,a_4\,\ldots)$.

Example

Newton’s Method gives rise to an algorithm in the sense of a process.

(NM) Start with a guess $x$ and let  $x:=x-f(x)/f'(x)$ (see colon-equals) repeatedly until either

• $f(x)$ gets sufficiently close to 0, in which case $x$ is the answer, or
• $f'(x)=0$, or
•  the process has gone on too long.

This algorithm is not hard to implement in C, Pascal, Python or many other programming languages.  But in all these cases you have to get the syntax exactly right and take care of a lot of details such as assigning labels to certain lines of the program. Note that (NM) is not any of those programs:  it is the process carried out by those programs.

Algorithms are discussed in Functions: images and metaphors and in detail in Wikipedia.

For many choices of starting point, Newton's algorithm will run forever and not give an answer. (In practice it would probably result in some kind of overflow and stop with an error message.) For many authors, "algorithm" means it always stops in a finite number of steps. For them, Newton's process is not an algorithm. There is a problem with their attitude: Determining whether a process is an algorithm is an undecidable problem.

alias

The symmetry of the square illustrated by the figure below can be described in two different ways.

• The corners of the square are relabeled, so that what was labeled $B$ is now labeled $A$. This is called the alias interpretation of the symmetry.  You are thinking of moving the labels around. "Honestly, officer, I am Albert Aardvark, not Bubba Bullfrog."
• The square is moved (turned), so that the corner labeled $A$ is now in the upper right instead of the upper left. This is the alibi interpretation of the symmetry.   You are thinking of moving the square together with its labels. "Honestly, officer, I was in Portland, Maine, not Portland, Oregon."

"Alibi" and "alias" are not mathematical properties of transformations;
they are ways to think about them. 

all

See universally true and universal conditional.

always

The statement "$x^2+1$ is always positive" (for a real variable $x$) is equivalent to the universally true statement, "For every real number $x$, $x^2+1$ is positive" (strictly greater than $0$.)

This use of "always" is noticeably less formal than other ways of expression universal statements. The mental image behind it is that you can vary $x$ all over the place as long as you want and the expression stays greater than $0$.

ambient

The word ambient is used to refer to a mathematical object, such as a space, that contains a given mathematical object. It is also commonly used to refer to an operation on the ambient space.

Example

"Let $A$ and $B$ be subspaces of a space $S$ and suppose $\phi$ is an ambient homeomorphism taking $A$ to $B$."  The point of this sentence is that $A$ and $B$ are not merely homeomorphic, but they are homeomorphic via the automorphism $\phi:S\to S$ of the space $S$.

and

any

arbitrary

Used to emphasize that there is no restriction on the mathematical structure referred to by the noun phrase that follows. In most cases the word adds no additional mathematical meaning to the statement. You may usually use "any" in this situation instead of "arbitrary".

Example

"The equation ${{x}^{r}}{{x}^{s}}={{x}^{r+s}}$ holds in an arbitrary group (or "in any group"), but the equation ${{x}^{r}}{{y}^{r}}={{(xy)}^{r}}$ requires commutativity."

In a phrase such as "Let S be an arbitrary set" the word arbitrary typically signals an expectation of an upcoming proof by universal generalization.

Consciousness raising about "arbitrary"

People new to abstract math may have a systematic tendency to underestimate how arbitrary a math object can be.  For example, the set $\{3,\,\,\pi ,\,\,\mathbb{R},\,\text{ the cosine function}\}$ is a perfectly good set.  It is arbitrary, and, I admit, weird, but it is a set.  Other examples:

More arbitrary sets.

The function in the Examples of Functions section called a finite function is an arbitrary function.

argument

This word has three common meanings in mathematical discourse.

• The angle a complex number makes with the real axis is called the argument of the number.
• The input to a function may be called the argument.
• A proof may be called an argument.

This word can cause cognitive dissonance.  In English,"argument" can mean either:

• Organized step by step reasoning to support a claim, as in,"The judge’s argument for finding the suspect innocent was based on the fourteenth amendment."
• The verbal expression of a disagreement, as in "George and Martha had an argument about the Venetian blinds."

The meaning of disagreement is the common one and it may carry a connotation of unpleasantness.  The three meanings in math that are given above have no connotation of unpleasantness.

assertion

An assertion in a mathematical text is a statement  in math English (more here) or the symbolic language (see symbolic assertion).  The author’s intention in writing an assertion may be difficult to determine (see intent).

assumption

An assumption is an assertion that is taken as true in a given block of text that is its scope. "Taken as true" means that any proof in the scope of the assumption may use the assumption to justify a claim without further argument.

Examples

• "Throughout this chapter, G will denote an arbitrary Abelian group."  In that chapter, a statement such as "The subgroup $B$ of $G$ is normal" can be taken to be true without further justification because every subgroup of an Abelian group is normal.
• A statement about a physical situation may be called an assumption.  Such a statement is then taken as true for the purposes of constructing a mathematical model.
• The hypothesis of a conditional assertion stands as an assumption during the statement of the assertion, and also through the proof if the proof is by modus ponens. 

at least, at most

For real numbers $x$ and $y$ the phrase"$x$ is at least $y$" means $x\ge y$.  The phrase " $x$ is at most $y$" means $x\le y$. 

Examples

• The statement "$B$ has at least one element" means exactly that $B$ is not empty.
• The statement "$B$ has at most one element" means $B$ is either the empty set or a singleton set (a set with exactly one element). (See also pattern recognition.)
• The statement "$n$ is at least $2$ and $n$ is at most $2$ " means $n=2$.
 

bar

A line drawn over a single symbol  is pronounced "bar". For example, $\overline{x}$ is pronounced "$x$ bar".  It often refers to the closure (in some sense) of a subset of a set.

Example

"Let $X$ be a topological space with subset $Y$, and let $\overline{Y}$be the closure of $Y$ in $X$."

Other names for this symbol are macron and vinculum.

be

The verb "to be" with its inflected forms "is" and "are" has many uses in the English language (see list in Wiktionary). Here I mention a few common usages in mathematical texts.

To assert that an object is an example of a certain type of object

For example,"Every integer is a rational number." 

Has a property

"Every integer is rational." This means the same thing as "Every integer is a rational number", but is expressed using different grammar. See property. 

To define a property

In defining a property, the word "is" may connect the definiendum to the name of the property, as in:

"A group is Abelian if $xy=yx$ for all elements $x$ and $y$."

Note that this is not an assertion that some group is Abelian, as in the previous entry; instead, it is saying what it means to be Abelian.  It is not always easy to know whether the author means such a statement as an assertion or as a definition.  In this example the fact that the word "Abelian" is in boldface communicates that it is a definition.  See definitions for more about this.

To define a type of object

In statements such as:

"A semigroup is a set with an associative multiplication defined on it."

the word "is" connects a definiendum with the condition defining it. Note again that it may not be clear whether this is an assertion or a definition.  See definitions for other examples.

Is identical to

The word "is" in the statement

"An idempotent function has the property that its image is its set of fixed points."

asserts that two mathematical descriptions ("its image" and "its set of fixed points") denote the same mathematical object. This is the same as the meaning of " =".

Asserting existence

For example, "There is a number $x$ which is bigger than $2$ ". See existential quantifier for other ways this can be worded.

binary operation

bracket

This word has several related usages.

Certain delimiters

In math texts, brackets can mean any of the delimiters parentheses, square brackets, braces, and angle brackets (see delimiters). Some American dictionaries and some mathematicians restrict the meaning to square brackets or angle brackets

Operation

The word "bracket" is used in various mathematical specialties as the name of an operator (for example, Lie bracket, Toda bracket, Poisson bracket) on an algebra. The operation called bracket may use square brackets, braces or angle brackets to denote the operation. The Lie bracket of $v$ and $w$ is always (as far as I know) denoted by $[v,w]$. On the other hand, notation for the Poisson and Toda brackets varies, with different authors using different symbols. 

Quantity

The word "bracket" may be used to denote the value of the expression inside a pair of brackets (in the sense of delimiters).  For example,

"If the expression \[\left(x^2-2x+1\right)+\left(e^{2x}-5\right)^3\] is zero, then the two brackets are opposite in sign." This usage may be obsolescent.

but

"and" with contrast

As a conjunction, but typically means the same as "and", with an indication that what follows is surprising or in contrast to what precedes it. This is a standard usage in English, not peculiar to mathematical English.

Example

" 5 is odd, but 6 is even."

introduces new property

Mathematical authors may begin a sentence with "but" to indicate that the subject under discussion has a relevant property that will now be mentioned.  For example, it may be relevant because it leads to the next step in the reasoning. The property may be one that is easy to deduce or one that has already been derived or assumed.

This usage may carry with it no thought of contrast or surprise.  Of course, in this usage "but" still means "and" as far as the logic goes; it is the connotations that are different.

Example

"We have now shown that $m = pq$, where $p$ and $q$ are primes. But that implies that $m$ is composite."

Example

(In a situation where we already know that $x=7$):

"... We have now proved that ${{x}^{2}}+{{y}^{2}}=100$. But $x$ is 7, so $y=\sqrt{51}$.

calculate

To calculate may mean to perform symbol manipulation on an expression, usually with the intent to arrive at another, perhaps more satisfactory, expression. "Compute" is used in much the same way. When mathematicians use one of these words, neither word necessarily indicates that the procedure was done on a computer or that a known algorithm was used. 

Examples

• "An easy calculation shows that the equation ${{x}^{3}}-5x=0$ factors into linear factors over the reals."  The calculation would probably go like this: "${{x}^{3}}-5x=x({{x}^{2}}-5)$, and we know that ${{x}^{2}}-5=(x-\sqrt{5})(x+\sqrt{5})$". "We know" is not something that would occur in a program or algorithm!
• "Let us compute the roots of the equation ${{x}^{2}}+3x-10=0$."  If you do this using the quadratic formula then that really is a computation using an algorithm.  But you could also try factoring $10=2\cdot 5$ and trying $(x-2)(x+5)$ and $(x+2)(x-5)$ – and the first one works.  This is not an algorithm but many mathematicians would still call it a computation or calculation.   (This remark about word usage is not based on lexicographical research, but only my own observations.)

call

Call may be used to form a definition.

Examples

• "A monoid is called a group if every element has an inverse."
• "Let $g=h^{-1}fh$. We call $g$ the conjugate of $f$ by $h$."
• "We call an integer even if it is divisible by $2$ ."

Note: Some object to using "call" with an adjective, as in the third example above.  However, this usage has been in the language for centuries.

case

Example

The Roman alphabet, the Greek alphabet, and the Cyrillic alphabet have two forms of letters, "capital" or uppercase, A, B, C, etc, and lowercase, a, b, c, etc.  

Case distinction always matters in mathematics.  It is common for example to use a capital letter to name a mathematical structure and the same letter in lowercase to name an element of the structure. This means that when you take notes at a lecture in abstract math, you must write "$S$" when the lecturer writes "$S$" and "$s$" when the lecturer writes "$s$"!

Other variations in font and style may also be significant.  See fraktur, boldface and blackboard bold.

cases

The word "cases" in this entry has no connection with the word "case" described in the preceding entry.

A concept is defined by cases if it is dependent on a parameter and the defining expression is different for different values of the parameter. This is also called a disjunctive definition or split definition.

Examples

• The absolute value function is defined by cases:\[|x|=\left\{ \begin{align}   & x\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(}x\geq0\text{)} \\  & -x\,\,\,\,\,\,\text{(}x\lt 0\text{)} \\ \end{align} \right.\]
• Let the function $f:\mathbb{R}\to \mathbb{R}$ be defined by
  \[f(x)=\left\{ \begin{align}   & x+1\,\,\,\,\,\,\text{(}x<2\text{)} \\  & x-1\,\,\,\,\,\,\text{(}x\ge 2\text{)} \\ \end{align} \right.\] Then for example, $f(1.9) = 2.9$ and $f(2.01) =1.01$.  This defines one function $f$, although the expression $f(x)$ is calculated using two different formulas depending on whether $x\lt 2$ or not.
• In many calculus books, the expression "$\sin^n$" means two different things depending on whether $n$ is $-1$ or some other integer. Details here.
• The word "if" in math English means two different things depending on whether it is used in a definition or in a theorem.

Disjunctive definitions may seem unnatural at first. This may be because real life definitions are rarely disjunctive. One exception is the concept of strike in baseball (swing and miss or didn’t swing when the ball was in the zone or one of several other possibilities).

category

The word category  is used with two unrelated meanings in math (Baire and Eilenberg-Mac Lane). It is used with still other meanings by linguists and cognitive scientists.

character

• A character in computing science is a letter or symbol in an alphabet, or its representation.  More here.
• A character in group theory is the trace of a linear representation.  More here.

check

closed under

A set is closed under an operation if the result of applying the operation to elements of the set is again an element of the set. 

Examples

• The set of natural numbers is closed under addition but not under subtraction.  (For example, 3 and 5 are natural numbers but 3 – 5 is not.)
• The set of real numbers $x$ for which $0\leq x\leq1$ (the unit interval) is closed under taking limits.
• The set of real numbers $x$ for which $0\lt x\lt1$ is not closed under limits. For example, $\lim_{n=1}^\infty\frac{1}{n}=0$ but $0$ is not in the set.

Much more about this idea is under closure in Wikipedia.

combination

An $r$-combination of a set $S$ is an $r$-element subset of $S$."Combination" is the word used in combinatorics. Everywhere else in mathematics, a subset is called a subset.

comma

And

A comma between symbolic assertions may denote and.

Example

The set $\{m|m={{n}^{2}},\,\,\,n\in \mathbb{Z}\}$ denotes the set of squares of integers. The defining condition is: "$m={{n}^{2}}$ and $n$ is an integer".

Note: The comma is used the same way in standard written English. Consider "A large, brown bear showed up at our tent".

Coreference

The comma may also be used to indicate a distributive constraint.

Example

"Let $x,\,\,y\ne 0$."  This means "Let $x\ne 0$ and $y\ne 0$", which means the same as "Let each of $x$ and $y$ be nonzero".

compute

See calculate.

constraint  

contain

If $A$ and $B$ are sets, the assertion "$A$ contains $B$" can mean one of two things:  $B\subseteq A$ or $B\in A$.  The sentence "$B$ is contained in $A$"  can mean either of these things as well. 

convention

coordinatewise

To apply a process coordinatewise to a list or vector is to apply the process to each coordinate individually and independently.

Examples

• If you negate $(1,3,-2)$ coordinatewise you get $(-1,-3,2)$.
• If you add $(1,3,-2)$ and $(-1,2,5)$ coordinatewise you get $(0,5,3)$.
• In the coordinatewise ordering of lists, $(a_1,a_2,a_3)\lt (b_1,b_2,b_3)$ if and only if all of $a_1\lt b_1$, $a_2\lt b_2$, and $a_3\lt b_3$ are true. This is entirely different from alphabetical ordering! ("Bed" comes before "cat".)

corollary

decreasing

define

Define means to give a definition. 

Examples

• "We define $n$ to be even if it is divisible by $2$."
• "Let us define $n$ to be even if it is divisible by $2$."
• "Define $n$ to be even if it is divisible by $2$."

Some students find the third form – define as a command – to be confusing: "What am I supposed to do when it tells me to define something?"  Answer:  You don’t do anything, you simply understand that from now on "even" means "divisible by $2$".

degenerate

The word degenerate is used in some disciplines to describe a particular math structure with one or both of the following properties:

a) Some parts of the structure that are normally distinct are the same in this particular example.

b) Some parameter is 0.  (Trivial is also sometimes used for this meaning.)

Examples

• A line segment is a degenerate isosceles triangle.  The two equal sides coincide and the third side has length 0.  This fits both a) and b).
• A critical point is degenerate if it satisfies a certain technical condition, namely that the determinant of its Hessian is 0.  This fits b).  A small perturbation turns a degenerate critical point into several critical points, so this also "sort of" fits a). 

Usage: The word degenerate is given a mathematical definition in some fields and is used informally in others.  Many specialties in math never use the word. 

More than half of the examples I turned up on JSTOR when doing research for the Handbook were refer­ences to degen­erate critical points.

denote

To say that an expression $E$ denotes a specific object $B$ means that $E$ refers to $B$; in an assertion containing a description of $B$, the description can be replaced by $E$ and the truth value of the assertion remains the same. $B$ is sometimes called the denotation of $E$.

Examples

• "The symbol $\pi$ denotes the ratio of the circumference of a circle to its diameter."
• "Let $f$ denote the function $x\mapsto x^2:\mathbb{R}\to\mathbb{R}$."

Opinions

• Some authors write sentences such as "Let $f$ denote any continuous function". "Denotes" is best restricted to referring to a specific object, so it is better to say, "Assume $f$ is any continuous function."
• Some authors write sentences such as "The ratio of the circumference of a circle to its diameter is denoted $\pi$." It is perhaps more transparent to say "denoted by $\pi$."

Acknowledgments: Atish Bagchi and Steven Krantz.

description

A description (in mathematical English and in the symbolic language) is a noun or noun phrase that refers to a particular object (a definite description) or an unspecified object of a certain type (indefinite description).

disjoint

Two sets $S$ and $T$ are disjoint if their intersection is empty, in other words if they have no elements in common.  A family of sets is pairwise disjoint if any two different sets in the family are disjoint.  

Examples

• ${1,2}$ and ${3,4,5}$ are disjoint.
• Let ${{\mathbb{R}}^{+}}$ denote the set of all positive real numbers and ${{\mathbb{R}}^{-}}$ the set of all negative real numbers.  Then ${{\mathbb{R}}^{+}}$and ${{\mathbb{R}}^{-}}$ are disjoint.  There is no real number that is both positive and negative.
• The set $\left\{ (n,n+1)|n\in \mathbb{R} \right\}$ is  a pairwise disjoint family of subsets of the reals (here $(n, n+1)$ denotes the open interval $\{x|\,n\lt x\lt n+1\}$, not the ordered pair.)
•  A statement such as "Let $A$, $B$ and $C$ be disjoint sets" usually means that the sets are pairwise disjoint.  When I was searching the literature for the Handbook, I could not find a clear example where such a statement meant that no element was in every set; it always meant no element was in any two different sets.

Warning

"Disjoint" requires two sets.  Don’t say things such as: "Each set in a partition is disjoint".   You could  say "each set in a partition is disjoint from each of the other sets."

distinct

When several new identifiers are introduced at once, the word distinct is used to require that no two different ones can have the same value.

Example

"Let $m$ and $n$ be distinct integers." This means that in the text that follows, one can assume that $m\neq n$.

Difficulties

Students may not understand that without a word such as "distinct", the variables may indeed have the same value. Thus "Let $m$ and $n$ be integers" allows $m = n$.

It is reported that E. H. Moore was sufficiently bothered by this phenomenon to say, "Let $m$ be an integer and let $n$ be an integer."

domain

A domain may be any of these:

• The domain of a function.
• A connected open set in a topological space. 
• A type of lattice studied in denotational semantics.
• A type of ring, more properly called an integral domain.

I recall as a graduate student being puzzled by the first two meanings given above, with the result that I spent a (mercifully short) time trying to prove that the domain of a continuous function had to be a connected open set.

e.g., i.e

The expression "e.g." means "for example". The expression "i.e." means "that is".  These abbreviations confuse many readers and writers, including writers in the research literature. They have are significantly different meanings. 

Examples

• "Let $n$ be an even integer, e.g. $6$."   This means that in the next bunch of text, $n$ might be $6$ but it doesn't have to be.
• "Let $n$ be an even integer, i.e. $6$."   This is bad.  Taken literally, it suggests that $6$ is the only even integer.
• "Let $n$ be an even integer, e.g. divisible by $2$ ."  Bad.  It says,"Let n be an even integer, for example divisible by $2$ ."  The reader expects the phrase after the comma to provide a particular integer that is even, but it doesn’t.    
• "Let $n$ be an even integer, i.e. divisible by $2$ ."   This one is good.

each

The word each may generally be used in the sasme way as all, every and any to form a universal quantifier.  However it is not common to see an assertion worded as "Each multiple of $4$ is even".  "Each" is more commonly used for a distributive plural.

Examples

• "For each even number $n$ there is an integer $k$ for which $n =2k$." Different numbers $n$ get different $k$'s (that's what "distributive plural" means). "Every" could be used here as well.
• "For each number $n$ there is an integer $k$ for which $n \lt k$." For this example, in contrast to the preceding one, there are many $k$'s that work for each $n$. However, there is no one number $k$ that works for every $n$.
• "Each student has a pencil." In ordinary non-mathematical English, this means that each student has a different pencil from any other student, but it does not rule out that some students have more than one pencil. This is different from the previous example, in which the same $k$ can work for many $n$'s. Thus the mathematical usage of "each" is not the same as the ordinary English usage.
• "Five students have two pencils each."  This means that each of the five students has two pencils (a different two for each student). This usage occurs in combinatorics, for example.  Some students do not grasp the significance of a postposited "each" as in this sentence.s

element

empty set

equals

equation

equipped

Used to associate the structure attached to a set to make up a mathematical structure. Also endowed.

Example

A semigroup is a set equipped with [endowed with] an associative binary operation.

equivalent

• Two assertions are equivalent (sometimes "logically equivalent") if they necessarily have the same truth values. See Equivalence of assertions.
• Two math objects may be equivalent by an equivalence relation. For example, $3$ and $-3$ are equivalent by the equivalence relation "has the same absolute value".

establish notation

exists

evaluate

factor

A factor is a constituent of a product.

Examples

• A factor of an integer is a divisor of the integer. For example, $6$ is a factor of $12$, since $12=6\times2$.
• You would not say $6$ is a factor of $9$, even though $9=6\times\frac{3}{2}$. There is a hidden convention that for integers, a factor of an integer must occur in a product consisting only of integers -- no fractions allowed.
• $x^2+1$ is a factor of the polynomial $(x^2+1)(x^2-1)$.
• You could also say that $x^2+1$ is a factor of the polynomial $x^4-1$. In this case, the presence of $x^2+1$ is hidden, but the statement is still correct.
• $8=3+5$, but $3$ is not a factor of $5$. The word factor refers only to products.

The word "product" is used for many other math objects besides integers and polynomials, and there are conventions about the use of "factor" in those cases as well. I have not done any lexicographical research about these other situations, but I know enough examples to assert that the situation is quite complicated. For example, a factor of a group $G$ does not have to occur in a product of groups that is isomorphic to $G$.

fallacy

A fallacy is an error in reasoning. Two fallacies with standard names that are commonly committed by students are affirming the consequent and denying the antecedent. See also argument by analogy.

Terminology The meaning of fallacy given here is that used in abmath in general. It is widely used with a looser meaning and often connotes deliberate deception, which is not intended here.

family

A family of sets sometimes means an indexed set (tuple) of sets (so differently indexed members may be the same) and sometimes merely a set of sets.

field

• A field is a function assigning values (usually denoting a physical quantity) to points in a space.
• A field is an algebraic structure allowing addition, subtraction, multiplication, and non-zero division subject to certain laws.

 The two meanings are unrelated, and you often have to tell from the context which one is meant. 

The word also refers to specialty, as in "My field is topological semigroups".  This meaning occurs in all academic subjects. "Field" is also discussed in the chapter on Names.

Used in problems in much the same way as give.

• A function $f$ fixes a point $p$ if $f(p) = p$. This is based on this metaphor: you fix an object if you make it hold still (she fixed a poster to the wall). This metaphor may not be helpful to Americans; in my observation, in the USA, the word nearly always means "repair".
• "Fix" is also used in sentences such as "In the following we fix a point $p$ one unit from the origin", which means that we will be talking about any point one unit from the origin (a variable point!) and we have established the notation $p$ to refer to that point. The metaphor behind this usage is that the value is "fixed" throughout the discussion; every reference to $p$ is to the same value, even though the value is not specified.

follow

Implies

The statement that an assertion $Q$ follows from an assertion $P$ means that $P$ implies $Q$ (if $P$, then $Q$).

Example

"The integer $n$  is divisible by 16.  It follows that it is even."

Grouping

The word "follow" is also used to indicate that some statements after the current one are to be grouped with the current one, or (as in "the following are equivalent)" are to be grouped with each other.

Example

"A set $G$ with a binary operation is a group if it satisfies the following axioms ... " This statement indicates that the axioms that follow are part of the definition currently in progress.

following are equivalent

The phrase "the following are equivalent" (or "TFAE") is used to assert the equivalence of the assertions that follow (presented in a list).

Example

The following are equivalent:

• "$x\geq0$ and $x\leq0$."
• "$x+1=1$".
• "$x=0$."

for all

See universal quantifier.

formula

• In informal usage in math, a formula is an expression allowing you to calculate some quantity.   More here.
• In math logic, a formula is an arrangement of symbols that represents a statement, possibly containing variables.   Formulas in this sense are almost always defined in terms of a particular formal language (MW, Wi).

The equation $A=\pi {{r}^{2}}$ is a formula in both senses, but $\pi {{r}^{2}}$("the formula for the area of a circle") is a formula only in the first sense above.  In mathematical logic, $\pi {{r}^{2}}$ would be called a term.

functional

forward reference

In the sentence, "When it is the subject of a clause, an indefinite description has the force of universal quantification", the pronoun "it" refers to the phrase "indefinite description", which occurs later in the sentence. This is called a forward reference (or "cataphoric reference"). Not only non-native English-speakers but many native English speakers do not automatically understand forward references. That is why in the entry for indefinite article, I reworded that sentence.

Another example: The question "Explain how to determine from the last digit of its octal notation whether an integer is even" baffled some of my discrete math students.

Forward reference is a type of coreference (see also "coreference" in the Handbook).

functional

generalization

give

Give is used in several ways in math English. often with the same sense it would be used in any academic text ("we give a proof... ","we give a construction ... ").

Produce a description of an example

To "give an object" means to describe it sufficiently that it is uniquely determined. A phrase of the form "give an $X$ such that $P$" means describe a object of type $X$ that satisfies predicate $P$. The word find can be used in this meaning as well.

Example

"Problem: Give (or find) a function of $x$ that is positive at $x$ = 0." Correct answers to this problem could be "the cosine function on the reals" or "the functions $f:\mathbb{R}\to \mathbb{R}$ given by $f(x)= 2x + 1$."

"Given" can mean "if" or "let"

$ $"Given sets $S$ and $T$, the intersection $S\cap T$ is the set of all objects that are elements of both $S$ and $T$."

This means "If $S$ and $T$ are sets, the intersection $S\cap T$ is the set of all objects that are elements of both $S$ and $T$" or "Let $S$ and $T$ be sets. Then the intersection $S\cap T$ is$\ldots$"

glue

To glue two math objects together is to identify them.  You make a Möbius strip by reversing the right edge of a rectangle and glueing it to the left edge, as described here.  This means you identify (in the mathematical sense) the reversed right edge with the left edge.  This is a metaphor that is also a literal description of what you do with a real rectangle made out of paper, too!

graph

The word graph has two unrelated meanings in undergraduate mathematics:

• The graph of a function.
• A (directed or undirected) graph is a structure consisting of nodes with directed or undirected edges that connect the nodes, often subject to further conditions that are discussed in detail in the Wikipedia article.

In both cases, the word graph is used both for the math object itself and also for a drawing of (often only part of) the math object.

hat

hold

An assertion $P$ containing a variable $x$ holds for $a$ if $P$ becomes true when $a$ is substituted for $x$. 

Examples

• "${{x}^{2}}+1\gt 0$ holds for all $x$."  
• "$\sin x=0$ holds for $x=\pi $."

• $\mathbf{I}$ (bold) may denote the unit interval, the set of real numbers $x$ for which $0\le x\le 1$.  Some authors use it to denote any bounded interval of real numbers.
• $I$ (not usually bold) may be used as the name of an arbitrary index set.
• For some authors, $\mathbf{I}$ or $\mathbb{I}$ (blackboard bold)  means the set of integers; however, $\mathbf{Z}$ or $\mathbb{Z}$ is also used for this.
• $\mathbf{I}$ may denote the identity matrix for a dimension given by context.

• $i$ (usually lowercase) denotes one of the square roots of –1.  (The other one is then $-i$).  In engineering, $ j$ is very frequently used instead of $i$ for this. 

These various meanings are quite commonly used without explanation.

i.e.

identify

To identify a math object $A$ with another object $B$ is to regard them as the same object.  In particular each element or point in $A$ is regarded as the same as a particular element or point in $B$.  In most cases, the author has in mind a particular way of matching the points in $A$ and the points in $B$.  This "particular way" may be presented informally or as a formal mathematical construction.

This is an informal or sug­ges­tive defi­nition of "identify", not a math defi­nition, because I haven't said what "ele­ment or point in" means.

The real plane

• You may regard each point of the real plane as having specific coordinates, an ordered pair of real numbers.  Example: "Consider the point with coordinates (2.3, -1.45)."
• On the other hand, you may think of the points of the real plane as identified with ordered pairs of real numbers.  Example: "Consider the point (2.3, -1.45)."

These are two different ways of thinking about the plane -- as a set of points represented by ordered pairs, or as a set of ordered pairs. This is an example of conceptual blending. 

Möbius strip

The Möbius strip may be constructed by identifying or gluing the edge \[\left\{ (x,0)|\,0\le x\le 1 \right\}\] of the unit square with the edge \[\left\{ (x,1)|\,0\le y\le 1 \right\}\] in such a way that $(x, 0)$ is identified with $(1 – x, 1)$. This can be thought of as saying that the coordinates  $(x, 0)$ and $(1 – x, 1)$ refer to the same point.  This is an example of identifying objects by a formal mathematical construction.

Usage: In ordinary English, "identify" means to give a name to. This presumably could cause cognitive dissonance.

identity

This word has three common meanings.

Equation that always holds

An identity in this sense is an equation that holds between two expres­sions for any valid values of the variables in the expressions. Thus, for real numbers (in fact for complex numbers), the equation  \[{{(x+1)}^{2}}={{x}^{2}}+2x+1\] is an identity. An identity in this sense may also be called a law.

Sometimes in the case of an identity the symbol "$\equiv$" is used instead of the equals sign.

Identity element of an algebraic structure

If $x\Delta e=e\Delta x=x$ for all elements $x$ in an algebraic structure with binary operation $\Delta$, then $e$ is an identity or identity element for the structure. Such an element is also called a unit element or unity.

A ring with identity may mean a ring that has an identity element, but it can also mean a ring subject to an algebraic identity.

Identity function

For a given set $S$, the function from $S$ to $S$ that takes every element of $S$ to itself is called the identity function.  (More about that here.) This is an example of a polymorphic definition. 

The identity function on $S$ is the identity element of the algebraic structure consisting of all functions from a set $S$ to itself with composition as operation.

if

In conditional assertions

The ways if is used in stating conditional assertions is discussed here.  See also let.

In definitions

It is a convention that, in a definition, the word if used to introduce the word being defined means "if and only if".

Example: (a) "An integer is even if it is divisible by $2$."

Some authors regularly use if and only if or "iff" in a definition, so that definition would be worded:

(b) "An integer is even if and only if it is divisible by $2$."

But note:  The following statement is not a definition, but an equivalence:

(c) "An integer is divisible by $2$ if and only if it is divisible by $–2$."

Even though "if and only if" is used in both (b) and (c), they don’t have the same status.   In statement (b) the left side and the right side are not symmetric.  The point of statement (b) is to give a meaning to the left side.  Statement (c) is symmetric; you could just as correctly say, "An integer is divisible by $–2$ if and only if it is divisible by $2$ ."

In the precondition of a definition

If can be used in the pre­condition of a definition to introduce some requirements on an object mentioned in the definition.  For example,  definition (a) above could have been worded this way:

"If $n$ is an integer, then it is even if it is divisible by $2$ ."

if and only if

If $P$ and $Q$ are two assertions, the statement "$P$ if and only if $Q$" is an assertion that $P$ and $Q$  are equivalent. That statement may be abbreviated "$P$ iff $Q$". 

imply

in

In is used in mathematical discourse in many of its normal English meanings, as well as in some meanings that are peculiar to mathematics.

• "$A$ is in $B$" can mean $A$ is an element of $B$ ($A\in B$).
• "$A$ is in $B$" can mean $A$ is a subset of $B$ ($A\subseteq B$).
• You may say "$A$ is in $B$" when $A$ is an equation whose solution set is included in $B$, or a geometric figure whose points are included in $B$. For example, "The unit circle ${{x}^{2}}+{{y}^{2}}=1$ is in the Euclidean plane."
• You may say "$A$ is in $B$" when $B$ is a symbolic expression and $A$ is part of $B$. Examples:
  • "$x$ is a variable in $3{{x}^{2}}+2x{{y}^{3}}$"
  • "$3{{x}^{2}}$ is a term in $3{{x}^{2}}+2x{{y}^{3}}$".
• "$A$ is $P$ in $B$", where $P$ is a property, may mean that $A$ has property $P$ with respect to $B$, where $B$ is a constituent of $A$ or a related structure (for example a containing structure). Examples:
  • "The set $\{x\in\mathbb{R}|\,0\leq x \text{ and } x\leq 1\}$ is topologically closed in $\mathbb{R}$."
  • If $A$ is a subgroup of the group $B$, then the statement "$A$ is normal in $B$" means that $A$ is a normal subgroup of $B$.
  • "$3{{x}^{2}}+2x{{y}^{3}}$ is differentiable in $x$". (It is also differentiable in $y$).
• You may describe an intersection using "in". For example, the sets $\{1,2, 3, 4\}$ and $\{1, 3, 5, 7\}$ intersect in $1$ and $3$.)

in general

• (a) A statement such as "$P$ is true in general" may mean that $P$ is true no matter what values you substitute for its variables. For example, "the equation ${{x}^{2}}-1=(x-1)(x+1)$ is true in general"  means that for all x, ${{x}^{2}}-1=(x-1)(x+1)$ is true. 
• (b) On the other hand, "In general, $P$" may sometimes mean only that there are some special cases where $P$ is true. Example: "In general, the Taylor series of an infinitely differentiable function has an infinite number of terms." (A polynomial is infinitely differentiable and is its own Taylor series, which therefore has only a finite number of terms.)
  The statement "there are some special cases" does not have a mathematical definition! The phrase just expresses the speaker's attitude toward the cases where $P$ is false.
• (c) Statements such as "In general, not $P$" probably mean that "$P$ is not necessarily true." Example: "In general, not every subgroup of a group is normal in the group"  means that an arbitrary subgroup of a group need not be normal in the group.  The statement specifically implies that there are examples where the subgroup is not normal in the group.

Notes

• In most circumstances, you can substitute generally for "in general".
• The Handbook article about "in general" has citations for usages (a) and (c), but not for (b), which I have some doubts about.

increasing

Let $f:\mathbb{R}\to \mathbb{R}$ be a function. 

• $f$ is increasing if for all $x$ and $y$, if $x\lt y$ then $f(x)\lt f(y)$.
• $f$ is decreasing if for all $x$ and $y$, if $x\lt y$ then $f(x)\gt f(y)$.
• $f$ is nonincreasing if for all $x$ and $y$, if $x\lt y$ then $f(x)\ge f(y)$.
• $f$ is nondecreasing if for all $x$ and $y$, if $x\lt y$ then $f(x)\le f(y)$.

Functions on ordered sets

These words "increasing", "decreasing" and so on can be used for functions to and from other ordered sets, as well.

Sequences

These words are also used for sequences, since a sequence is a function on its index set.   For example, a sequence ${{a}_{1}},\,\,{{a}_{2}},\,\,{{a}_{3}},\,\ldots.$ is increasing if for all $i$ and $j$ in its index set, if $i\lt j$ then ${{a}_{i}}\lt{{a}_{j}}$.

inequality

An inequality is an assertion of the form $s\,\alpha \,t$, where $s$ and $t$ are terms and $\alpha$ is one of the relations $\lt$, $\gt$, $\leq $ or $\geq $.

Examples

The assertions \[(4+3)^2\gt 8\times 6\] and \[\sin \frac{3\pi}{2}\leq0\] are inequalities.

If you are bothered by the second assertion, see unnecessarily weak assertions.

The word "inequality" is not usually used
to mean an assertion of the form "$s\neq t$".

inhabits

The phrase "$A$ inhabits $B$" can mean any of several things:

• $A$ is an element of the set $B$.
• $A$ is an entry in the list $B$.
• $A$ is an expression in the delimiters $B$.  (For example,"The function inhabiting the curly braces is increasing.")

Lives in is used similarly in conversation, at least for "element of", but I have found only a few citations in print.

injective

Functions can have a property called injective, and so can modules.  These two uses are unrelated.

input

If $f(x)=x^2+1$, then $f(2)=5$.

• In this example, $2$ is the input and $5$ is the output.
• If the input is $3$, the output is $10$.
• The input is also called the argument, and the output is the corresponding value.

integral

This word has three different meanings.

Being an integer

Integral is used as an adjective to require that the noun phrase it modifies denote an integer. For example, $8$ is an integral power of $2$ but $9$ is not.  This is also used for the more general notion of algebraic integer (MW, Wi).

Antiderivative

An integral of a function is an antiderivative of the function.  It may be called the indefinite integral to distinguish it from the definite integral mentioned below.  The indefinite integral of $f$ may be denoted by $\int{f(x)\,dx}$ which determines the integral only up to a constant.

You may refer to a specific antiderivative using the form $\int_{c}^{t}{f(x)\,dx}$.

The word integral is also used to denote a solution of a more general differential equation.

Definite integral

Integral is also used to denote a definite integral: this operator takes an integrable function and an interval (or more general space) on which the function is defined and produces a number. The definite integral of $f$ on an interval $ [a, b] $ is denoted $\int_{a}^{b}{f(x)\,\,dx}$ or $\int_{a}^{b}{f}$.  

intuition

Mathematicians may use the word intuition in referring to some image or metaphor they have about a type of object. 

Examples

• "Intuitively, the squaring function is a machine that turns, for example, $3$ into $9$."
• They may admit the intuition but point out its weaknesses: "Intuitively, an open set in the plane is every point in a region not on its boundary.  In fact, not all open sets fit this description, for example…" 

See also the discussion of intuition in Diagnostic examples.

is

See be.

just

One use of the word just in mathematical discourse is to indicate that what precedes satisfies the statement that occurs after the word "just".  

Example

(Assuming $r$ and $s$ are known to be integers greater than $1$).

"... Then $m = rs$. But that just means that $m$ is composite".)

Notes

In such sentences, "just" may commonly be omitted without changing the meaning, but in many cases it probably shouldn’t be omitted because it is an important indicator of the logical flow.

Before I did the research (finding citations of usage) for the article on "just" in the Handbook, my own perception of this usage was that the word "just" meant that what followed was equivalent to what preceded.  That’s a small example of the vital necessity of checking citations before making pronouncements about math English!

just in case

This phrase means that what follows is logically equivalent  to what precedes.  A search of Jstor reveals that in math research articles it is used mostly but not entirely by logicians.

Example

"An integer is even just in case it is divisible by $2$ "

In ordinary English (used more in Britain than in the USA, I think), "just in case" means "to guard against the possibility of".   

Example

"I will take an umbrella just in case it rains" means something like "I will take an umbrella in order to guard against the (undesired) possibility that it might rain."

This can cause cognitive dissonance: The statement "An integer is even just in case it is divisible by $2$ " certainly does not mean "An integer is even in order to guard against the possibility that it might be divisible by $2$ ."

Acknowledgments: Thanks to suggestions from Geoffrey K. Pullum.  See his Language Log entry concerning this phrase.

large

• Large said of a number can mean large positive or large negative, in other words large in absolute value.
• A text that says one set is larger than another may be referring to the ordering by inclusion or to cardinality.  For example, the set of integers is larger than the set of positive integers in the inclusion ordering but not in cardinality, whereas {1, 2, 3, 4} is larger than {2, 5, 6} in cardinality but not in the inclusion ordering.

lemma

A theorem.  Usually when an author calls a theorem a lemma, the connotation is that the lemma is not of interest for itself, but is useful in proving other theorems. However, some lemmas (König's Lemma (MW, Wi), Schanuel’s Lemma (PlanetMath), Zorn's Lemma (MW, Wi)) have become quite famous and are explicitly taught in some courses because of their usefulness.

In old books sometimes the plural of lemma is lemmata.

let  

"Let" is used in several different ways in mathematical English.

Introducing a new symbol or name

The most common use of "let" is to introduce a new symbol or name.  This makes it a kind of definition.  The scope is usually restricted to the current section of text.  In contrast, the scope of a formal definition explicitly using the word "definition" is generally the whole discourse.

There is no logical distinction between this use of "let" and a formal definition.  The difference apparently concerns whether the newly introduced expression is for temporary use or meant to hold throughout the text, and perhaps whether it is regarded as important or not.  

Example

Consider the theorem

"An integer divisible by $4$ is divisible by $2$." A proof could begin this way: "Let $n$ be an integer divisible by $4$." This introduces a new variable symbol $n$  and constrains it to be divisible by $4$.

Example

Suppose the theorem of the preceding example had been stated this way:

"Let $n$ be an integer. If $n$ is divisible by $4$ then it is divisible by $2$." Then the proof could begin, "Let $n$ be divisible by $4$." In this sentence, $n$ is introduced in the theorem and is further constrained in the proof.

These two examples illustrate that whether a new symbol is introduced or a previous symbol is given a new interpretation is a minor matter of wording; the underlying logical structure of the argument is the same.

To consider successive cases

Example

"Let $n \gt 0$.... Now let $n\le 0$" If, assume and suppose seem to be more common that "let" in this use.

To introduce an arbitrary object

To pick an unrestricted object from a collection with the purpose of proving an assertion about all elements in the collection using universal generalization. Often used with arbitrary. If, assume and suppose can be used here.

Example

If you are writing a proof of a theorem that claims something is true about all prime numbers, your proof might start out: "Let $n$ be a prime number".  During the proof you can use the fact that $n$ is a prime number, but you may not make other special assumptions about it.

To name a witness

To provide a word or symbol for an arbitrary object from a nonempty collection of objects.   Equivalently, to choose a witness to an existential assertion that is known to be true.  If, assume and suppose can be used here.

Example

In proving a theorem about a differentiable function that is increasing on some interval and decreasing on some other interval, you might write:

"Let $a$ and $b$ be real numbers for which $f'(a)\gt0$ and $f'(b)\lt0$."

These numbers exist by hypothesis.

Example

In the context that $G$ is known to be a noncommutative group:

"Let $x$ and $y$ be elements of $G$ for which $xy\neq yx$.

The following is a more explicit version of the same assertion:

"Let the noncommutative group $G$ be given. Since $G$ is noncommutative, the collection $\left\{ (x,y)\in G\times G|\,xy\ne yx \right\}$ is nonempty, so we may choose a member ($x$,$y$) of this set..."

Example

In proving a function $F:S\to T$ is injective, you might begin with "Let $x,\,\,x'\in S$ be elements for which $F(x)=F(x')$".  These elements must exist if $F$ is non-injective: in other words, this begins a proof by contrapositive. The existence statement for which the elements $x,\,\,x'\in S$ are witnesses is implied by the assumption that $F$ is not injective.

"Let" meaning "define"

Let can be used in the defining phrase of a definition.

Example

"Let an integer be even if it is divisible by 2."

This usage strikes me as unidiomatic. It sounds like a translation of a French ("Soit...") or German ("Sei...") subjunctive. If, assume and suppose cannot be used here.

Other words can replace "let" -- sometimes

Assume, define, if, and suppose can be used instead of "let”. The syntax varies depending on which word is used, and some of these words cannot be used in certain situations. There are other subtle differences as well.

Example

You can say,

• "Let $X$ be $\ldots$ ”, but you have to say
• "Assume [Suppose] $x$ is $\ldots$ ”

Example

• "If $x = 1$” cannot be a complete sentence, but
• "Let $x = 1$” is a complete sentence.

Examples:

All these sentences have the same mathematical content.

• "Let $S$ be an odd integer.  Then $S$ is not divisible by $2$".
• "Assume $S$ is an odd integer.  Then $S$ is not divisible by $2$".
• "Suppose $S$ is an odd integer.  Then $S$ is not divisible by $2$".
• "If $S$ is an odd integer, then $S$ is not divisible by $2$."

The periods after "integer" in the first three sentences may be replaced by a semicolon, but not by a comma.  The comma in the fourth sentence cannot be replaced by a period; "If $S$ is an odd integer" is not a complete sentence.

Acknowledgments: Atish Bagchi.

linear

• a) A function $f:\mathbb{R}\to \mathbb{R}$ is linear if it is of the form $f(x)=ax+b$, where $a$ and $b$  are real numbers.  The graph of such a function is a straight line.
• b) If $M$ and $N$ are modules over a ring $A$, then a map $f:M\to N$ is linear (or an $A$-linear map) if for all $a$ and $a'$  in $A$ and $m$ and $m's$  in $M$, $f(am+a'm')=af(m)+a'f(m')$.

Note: An $F$ vector space over a field $F$ is a module over $F$.  In particular, $\mathbb{R}$ is a vector space over itself, so the two definitions of linear for functions $f:\mathbb{R}\to \mathbb{R}$ conflict. In particular, an example of (a) with $b\neq0$ is not an example of (b).  This conflict may be avoided by referring to functions satisfying (a) as affine.  But it might be a problem explaining to a high school student that a function whose graph is a straight line is not necessarily linear.

The word "affine" is generalized to vector spaces: see MathWorld.

Acknowledgments: Mariana Montiel.

logarithm

For real numbers $a$ and $x$ with $a$ positive and not equal to 1,  the expression ${{\log }_{a}}x$ denotes the number $y$ for which ${{a}^{y}}=x$.   The number $a$ is called the base. 

You will quite often see the expression "log $x$" with the base omitted (so that the expression has a suppressed parameter.)  This expression can mean three different things, depending on the specialty of the text. 

• In pure math research articles and books, the base is usually e.  This is called the natural logarithm.  In calculus texts it may be written "ln".
• In texts by scientists and in many modern calculus texts, the base is likely to be $10$.
• In computing science, the base is likely to be $2$ and may be written "lg".

mathematical logic

map                                                  

maximize

To maximize a function is to find values of its argument for which the function has a maximum. Minimize is used similarly.

Example

The number $t=5$ maximizes the function $h(t)=25-{{(t-5)}^{2}}$ (discussed here.)

The metaphor behind this usage seems to be: vary the input until you find the largest value.  This is the way most functions are maximized or minimized, using more or less sophisticated methods of varying the input – the bisection method and Newton’s Method are examples of this.

mean  

To form a definition  

"Mean" may be used in forming a definition.

Example

"To say that an integer is even means that it is divisible by 2."

Implies

To say that an assertion $P$ means an assertion $Q$ may signify that implies $Q$.

Example

"We have proved that $4$ divides $n$. This means in particular that n is even."

Math authors do not always make it explicit whether they are using "mean" to give a definition or to describe an implication.

Average

Mean is also a technical term, referring to the arithmetic average.

minus

The word minus can refer to both the binary operation on numbers, as in the expression $a-b$, and the unary operation of taking the negative: negating $b$ gives $-b$. In current usage in American high schools, $a – b$ would be pronounced "a minus b", but $–b$ would be pronounced "negative $b$". The older usage for $-b$ was "minus $b$" and many old fuddy duddy college teachers like me forget and call it "minus $b$" sometimes. 

Don’t assume a minus sign before an expression makes it negative.  The expression "$–t$" denotes a positive number  if $t$ happens to be $–42$.   Don’t be misled by the fact that we call it "negative $t$".  See also subtraction.

modulo

The phrase "$x$ is the same as $y$ modulo $E$" means that $x$ and $y$ are elements of some set, $E$ is an equivalence relation on the set, and $x\, E\, y$. There are variants to this usage:

• $x$ is the same as $y$ mod $E$.
• $x = y$ modulo $ E$ ( or mod $E$).
• $x \equiv y$ modulo $ E$ ( or mod $E$).
• $x$ is the same as $y$ up to $E$.

The word arose from a special equivalence relation in number theory, which has two conflicting special notations in pure math and computing science that I have seen cause confusion among students.

For integers $a$, $b$ and $n$, the expression $a\equiv b\bmod n$ means that $a – b$ is divisible by $n$.  (This relation is an equivalence relation. For example, $23\equiv 11\bmod 3$ and $-4\equiv 16\bmod 5$.  In this usage, the symbol "mod" occurs as part of a three-place assertion.

In this usage in number theory, the symbol "$\equiv$" does not mean "identical to".

Mod in computer languages

In most computer languages and in computing science texts, the expression "$a\bmod n$" means the least nonnegative remainder obtained when $a$ is divided by $n$.  For example, $23 \bmod 3 = 2$ and $-4 \bmod 5 = 1$.  (See the Wikipedia article on modulo for details.)  In this usage, mod is a binary operation on the integers.

Colloquial mod

People in math-related fields use "mod" in phrases such as "The administration kept my salary the same modulo [or mod] inflation". Presumably the equivalence relation here is something like: "One dollar in 2002 is equivalent to $1.02 in 2003."

In particular, a statement such as "The alternating group on three letters is the same as the cyclic group of order $3$ up to isomorphism" (less often, "modulo isomorphism") refers to the equivalence relation of two structures being isomorphic.  

model

multiplication

The word multiplication is the name for many binary operations.

• Multiplication of numbers: The product of $3$ and $5$ is $15$, typically denoted by $3\times5$. (See Common arithmetic operations.)
• Multiplication of variables in algebra, typically denoted by juxtaposition; for example, the product of $a$ and $b$ is denoted by $ab$.
• Many structures studied in undergraduate math courses involve a binary operation that is typically called multiplication and denoted by juxtaposition. For example, the product of two elements $a$ and $b$ of a group may be denoted by $ab$.

A trap many students fall into

Multiplication of numbers is commutative: For any numbers $a$ and $b$, $ab=ba$. This is not necessarily true in groups. For example, let $a$ be the permutation of the set $\{1,2,3\}$ that switches $1$ and $2$ and let $b$ be the permutation of the set $\{1,2,3\}$ that switches $2$ and $3$. Then the product $ab$ (do $a$ then $b$) takes $1$ to $3$, $3$ to $2$ and $2$ to $1$, whereas the product $ba$ takes $1$ to $2$, $2$ to $3$ and $3$ to $1$.

Some group theory texts write "$ab$" to mean "do $b$ then $a$". Some of them are even nice enough to tell you which way they are doing it.

Countless legions of group theory students have quite unconsciously tried to prove some theorem by writing (for example) $aba=a^2b$ and similar blunders.

The definition of "group" does not require that multiplication be commutative, and:

If a math definition defines a word
that is familiar to you in some other context,
you can't use properties it has in that context in a proof.

multivalued function

must

You often see "must be" used in math English when "is" would give the same meaning. It is used with verbs other than "be" in the same way. This may be to emphasize that the fact being asserted can be proved from facts known in the context of the discussion.

Examples

• "An odd prime must be greater than $2$."
• "Let $C=\left\{ 1,\,\,2,\,3 \right\}$.  If $C\subseteq A\cup B$, then one of $A$ and $B$ must contain at least two elements of $C$."

Other uses of "must" in mathematical discourse are generally examples of the way the words is used in ordinary discourse.

namely

Used to indicate that what follows is one of the following:

• A definition or more detailed description of what precedes.  
• The particular example of what precedes that the author is going to talk about.

Examples

• "Let $G$ be an Abelian group, namely a group whose multiplication is commutative."   What comes after "namely" is simply the definition of "Abelian group".
• "We now consider a specific group, namely ${{S}_{3}}$".   Here, "${{S}_{3}}$" identifies the particular object alluded to by "a specific group".
• "$12$ has two prime factors, namely $2$ and $3$."    The statement "$12$ has two prime factors" claims the existence of one or more objects, which are in fact $2$ and $3$. 

Note: A statement such as "$210$ has several prime factors, namely $2$ and $3$" sounds incorrect to me, since $210$ also has $5$ and $7$ as prime factors.  I would use "for example" or "including" instead of "namely" here. 

natural number

necessary

$Q$ is necessary for $P$ if $P$ implies $Q$, in other words if the conditional assertion "If $P$, then $Q$" is true.  Examples are given under conditional assertion.

The motivation for the word "necessary" is that the assertion "$P$ implies $Q$" is logically equivalent to "not $Q$ implies not $P$" (see contrapositive), so that for $P$ to be true it is necessary in the usual sense of the word for $Q$ to be true.

never

A statement like "$A$ is never $P$" is used when $A$ depends on a variable (parameter) $x$ and $P$ is a property; it means "For every value of $x$, $A$ does not have property $P$".

Examples

• "A real number never has a negative square.” The indefinite article means that this can be reworded as "If $x$ is any real number, then $x^2\geq0$.
• "The sine function is never greater than $1$” means that for all (real) $x$, $\sin x\leq1$. A strict writer would have written, "The value of the sine function is never greater than $1$". See Naming a function by its value.

nonincreasing

not

See negation.

nonnegative

A real number $r$ is nonnegative if it is not negative, in other words if $r\ge 0$.

nontrivial

See trivial.

notation

Notation is a system of signs and symbols used as a representation of something.  The word "notation" may be used to refer to a whole system or to a way of representing a particular object.

Examples

• The symbolic language of mathematics is a system of notation. 
• The standard notation for music is a system of notation.
• The usual notation for the empty set is "$\varnothing $", although some writers misunderstand it and use the Greek letter $\phi$ (phi).
• The notation $\{1, 2, 3\}$ refers to the set whose elements are exactly $1$, $2$ and $3$. 

Notes

• It would likely be better to say "symbol" instead of "notation" when the notation is a single symbol, such as $\varnothing $.
• The notation for a math object, if it is simple enough, may be a particular example of an image we associate with the object.  More here.
• The same math object may be referred to by many different notations (more here).

Establish notation

Mathematicians may say, "Let's establish some notation",  meaning they will introduce a specific combination of certain symbols to refer to a particular mathematical object. This is a type of definition on the fly, so to speak. See also fix and let.

In this text, the author supposes $x\notin A$ presumably because they are doing a proof by cases.

Bring up a fact that is needed

"Now” may be used to point out a fact that is already known or easily deduced and that will be used in the next step of the proof.

Example 2

In a situation where we already know that $x=7$, one could say, "We have that $x^2+y^2=100$. Now, $x=7$, so $y=\sqrt{5}$." This is similar to the second meaning given for but.

Here

"Now” may simply refer to the point in the text at which it occurs.

Example

"We are now in a position to give a conceptual proof of Pringsheim's Theorem."

The metaphor here is that the reader has been reading straight through the text (unlike a grasshopper) and at the current moment she sees this word "now”. As such it does not really add anything to the meaning.

The Handbook article on "now" includes several more examples.

Acknowledgments

Atish Bagchi.

A theorem is said to be obvious if the speaker’s mental representation of the mathematical object mentioned in the theorem makes the truth of the fact immediately perceivable. "Obvious" should be used carefully; you need to be sure the listener shares your particular mental representation!   See also trivial.

only if

operator

or

order

The order of a mathematical object is a number (or other math object) associated with that object.  What "order" means depends on the mathematical specialty. 

Examples

• The order of a differential equation is the highest derivative occurring in the equation.  For example the order of the equation $u''-2u'=u$ is $2$.  Warning:  The order of $u''-2u'=3{{u}^{17}}$ is also $2$.  
• A very common meaning for order is the cardinality of some set associated with the structure.  For example, the order of a group is the number of elements in the group. 
• An element of a group also has an order, which is the order in the preceding sense of the cyclic subgroup it generates (it may be infinite).
• Degree is frequently used similarly.  Some structures, for example permutation groups, have both an order and a degree, and those two words mean different things.
• Order can also refer to an ordering, as in "Consider the usual order on the reals."
• See also rank.

ordered pair

• For any math objects $a$ and $b$, the ordered pair $(a,b)$ is a math object determined completely by the fact that its first coordinate is $a$ and its second coordinate is $b$. (This is a specification of "ordered pair", not a definition.)
• The operative principle of ordered pairs is: $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$. This is all you need to know about ordered pairs.
• For example, $(3,5)$ is the ordered pair with first coordinate $3$ and second coordinate $5$. It is not the same as the ordered pair $(5,3)$. In contrast, the set $\{3,5\}$ is the same set as $\{5,3\}$.
• The entries in $(a,b)$ can be any math object, not just numbers. For example, the ordered pair $(\mathbb{R},42)$ has first coordinate the set of all real numbers and the second coordinate the number $42$. Yes, this is a ridiculous example. But it shows the generality of the concept of "ordered pair".

pairwise disjoint

partial function

parameter

partition

The word partition has several distinct meanings in math. 

• A partition of a set into disjoint blocks (same as an equivalence relation).
• A partition of an integer, expressing the integer as a sum of integers.
• A partition of an interval, used in the theory of integration.
• A partition of unity of a topological space.

All of these ideas involve the concept of breaking something up into distinct pieces, but they are genuinely distinct ideas. 

permutation

Permutation is defined in the literature in two different ways:

• A permutation of an $n$-element set is a sequence of length $n$ in which each element of the set appears once.  This is discussed in detail in Mathworld.

• A permutation of a set is a bijection from the set to itself.

Of course, the two definitions can be converted into each other, but they give different mental images. The discussion in Wikipedia gives both definitions.

pons asinorum

positive

In most texts and university courses, the phrase " $x$ is positive" means $x \gt 0$. In a few texts and in France (and possibly other places in Europe), it may mean $x\ge 0$.

Acknowledgment: The members of the RUME discussion list.

power

Here are three variant phrases that say that $125=5^3$:

• "$125$ is a power of $5$ with exponent $3$".
• "$125$ is the third power of $5$".
• "$125$ is $5$ to the third power".

Some students are confused by such statements, and conclude that $3$ is the "power". This usage appears in print in Wikipedia in its entry on Exponentiation (as it was on 22 November 2016):

...$b^n$ is the product of multiplying $n$ bases: \[b^n = \underbrace{b \times \cdots \times b}_n\] In that case, $b^n$ is called the $n$-th power of $b$, or $b$ raised to the power $n$.

As a result, students (and many mathematicians) refer to $n$ as the "power" in any expression of the form "$a^n$". The number $n$ should be called the "exponent". The word "power" should refer only to the result $a^n$. I know mathematical terminology is pretty chaotic, but it is silly to refer both to $n$ and to $a^n$ as the "power".

Almost as silly as using $(a,b)$ to refer to an open interval, an ordered pair and the GCD. (See The notation $(a,b)$.)

Suggestion for lexicographical research: How widespread does referring to $n$ as the "power" come up in math textbooks or papers? (See usage.)

Thanks to Tomaz Cedilnik for comments on the first version of this entry.

prime 

A prime is a typographical symbol, and also a positive integer that has exactly two positive integer divisors (Wi, MW). 

proper

• If a subset $T$ of a set $S$ is not $S$, then it is a proper subset of $S$. (See also here.)  This is also used with substructures of a structure (proper subgroup, and so on).  
• Some authors require that a proper subset of $S$ be not only not all of $S$ but also not be empty.  Unless the author makes it clear, you cannot tell which is meant. 
• The word proper is also used by some authors to mean nontrivial; for example, a proper automorphism would be a non-identity automorphism.

proposition

This word is used in conflicting ways in mathematical writing and in mathematical logic.

• In mathematical writing, proposition is another name for theorem, a statement that has been proved true.

• In mathematical logic, "proposition" is common terminology for a  sentence in math English or the symbolic language that is either true or false. In this website I use the word statement for this meaning of proposition.

quotient

The word quotient is used with many different meanings in math, all of which are closely related.  The Wikipedia article covers a lot of these meanings.

range

The range of a function may mean its codomain or its image.  Both usages are common in undergraduate texts.

rank

The property of rank is defined for many different kinds of math structures. The word generally means different things for different structures. The Wikipedia article covers many of these meanings.

Group, abelian group, permutation and permutation group each has a different meaning of "rank", and graph has at least two unrelated meanings of "rank".

recall

Used before giving a mathematical fact (definition, theorem or proof).

Example

"Recall that an integer is even if it is divisible by 2."

The intent seems to be that the author expects that you already know the meaning of the defined term, but just in case here is a polite reminder.

I often see "recall that" when I don't recall anything of the sort.

result

• The value produced by a function at a given input may be called the result of the function at that input.
• A mathematical fact that has been proved (a theorem) may also be called a result.

revise

In the United States, to revise a document means to change it, hopefully improving it in the process. Speakers influenced by British English use "revise" to mean what Americans mean by "review".  In particular, students may talk about "revising" for an upcoming test. In this case there is no implication that anything will (or will not) be changed.

Note: This entry has nothing directly to do with math or math English, but I have several times witnessed the confusion it can cause in academic circles and so thought it worth including here.

root

• A root of an equation $f$($x$) = 0 is a value c for which $f$($c$) = 0.
• This value $c$ is also called a a zero of the function $f$.
• The $n$th root of a number $a$ is a number $c$ for which ${{c}^{n}}=a$; in other words, $c$ is a root of the equation ${{x}^{n}}-a=0$.

Usage: Some say it is incorrect to refer $c$ as a "root of $f$" instead of "zero of $f$".  In fact, using "root" in this situation appears to me to be more common that using "zero",  particularly when the function is a polynomial.  Mathematica uses "root" in this case, too.

round parentheses

satisfy

Examples

• The number $r$ satisfies the equation $3x+2=8$ if the equation $3r+2=8$ is correct. So $2$ satisfies the equation but for example $3$ does not.
• The equation $x^2=2$ is satisfied by two numbers, namely $\sqrt{2}$ and $-\sqrt{2}$. No other numbers satisfy the equation.
• Infinitely many numbers $x$ and $y$ satisfy the equation $3x+5y=2$, for example $(-1,1)$, $(2,-1)$ and $(3,-\frac{7}{5})$.
• $5$, $42$ and many other numbers satisfy the inequality $x\geq5$, but $-42$ and $4$ do note.
• No real number satisfies the inequality $x^2\lt0$.

sequence

sign

• The word sign is used to refer to the symbols "$+$" (the plus sign) and "$-$" (the minus sign).
• The word "sign" is also sometimes used to refer to other symbols, for example "the integral sign".
• It is also used to refer to the question of whether an expression represents a real number that is positive or negative. 

Example

If $f(x)={{x}^{2}}$, you may say, "For negative $x$, $f$($x$) and $f$'($x$) are opposite in sign." Observe that for negative $x$, the expression $f'(x)$ denotes a negative number even though the expression has no negative sign in it.

some

subscript

An expression such as ${{a}_{2}}$ commonly means that $a$ is the entry indexed by $2$ in a sequence.  Since a sequence is a function on its index set, authors may refer to ${{a}_{i}}$ as a "function on $I$", and use language such as"$a$ is increasing in $I$". 

subtract

Removing objects from a collection

In ordinary English, if you subtract from a collection you make it smaller, and if you add to a collection you make it bigger.

Applying the operation of subtraction

When $a$ and $b$ are numbers, to subtract $b$ from $a$ means to add $-b$ to $a$, written $a-b$. This process can clash with the ordinary English meaning of "subtract".

• If you subtract $3$ from $5$ you get $2$, which is smaller than $5$. No problem.
• But if you subtract $-3$ from $5$ you get $8$, which is bigger.

This difference in the mental image of subtraction can may you think, for example, that $x-y$ must be smaller that $x$. But that is wrong if $y$ is negative!

Both these usages (adding to a collection and applying the operation of addition) occur in mathematical writing.

See also minus and subtraction.

such that

For an assertion $P$, a phrase of the form "$c$ such that $P(c)$" means that the assertion $P$ must be true of $c$.

Examples

• "Let $n$ be an integer such that $n\gt 2$" means that in any following assertions that refer to $n$, one can assume that $n\gt 2$.
• "The set of integers $n$ such that $n\gt 2$" refers to the set  $\left\{ n|\,n\gt 2 \right\}$.  (See setbuilder notation and the.)

Notes

• In pronouncing $\exists x\,P(x)$ the phrase "such that" is usually inserted. This is not done for the universal quantifier.  So "$\exists x(x\gt 0)$" is pronounced "There is an $x$ such that $x$ is greater than $0$", but " $\forall x(x\gt 0)$" is pronounced "For all $x$, $x$ is greater than $0$".
• Yes, I know that "$\forall x(x\gt 0)$" is false.

sufficient

$P$ is sufficient for $Q$ if the statement "If $P$, then $Q$" is true (see conditional assertion.)   You can also say $P$ suffices for $Q$. The idea behind the word is that to know that $Q$ is true it is enough to know that $P$ is true.

superscript 

• An expression such as ${{a}^{n}}$ may denote $a$ to the $n$th power.
• An expression such as "${{a}^4}$" may also indicate that it is the $4$th entry in a sequence indexed by a superscript.  This is common in tensor notation, for example. 

symbol

As used in math, a symbol, is either

• A typographical character, such as "$x$", "$\phi $", "$\bigcup{{}}$" and $\infty $.
• An expression that stands for something. For example, the individual symbols just listed each will have a meaning in a math text (not always the same meaning). The expression can also be compound; for example the Legendre Symbol has the form ($a$/$p$), where $a$ is an integer and $p$ is a prime number.  

These two meanings are genuinely different. The symbol $\phi$ has two distinct meanings in these sentences:

• "$\phi$" is pronounced "fee".
• $\phi = \frac{1+\sqrt{5}}{2}$

It is best if an author puts quotes around the symbol when they mean the typographical symbol, but they don't always do that. See Mathworld and Wikipedia for much more about symbols.

symbol manipulation

Symbol manipulation  is the use of algebraic rules, or rules of some other computational system, to change a symbolic expression to an equivalent one.  For example, you can change $3{{x}^{2}}+3$ to the equivalent expression $3({{x}^{2}}+1)$ using the distributive law for algebra, and you can change $\int_{1}^{3}{{{x}^{2}}+{{x}^{3\,}}\,dx}$ into $\int_{1}^{3}{{{x}^{2}}\,dx}+\int_{1}^{3}{{{x}^{3}}\,dx}$ using a rule of integration.

symbolic logic

T

tangent

The word tangent may refer either 

• to a straight line in a certain relation to a differentiable curve, or
• to the trig function tangent.

These two meanings are related: the trigonometric tangent of an angle $\theta$ is by definition the length of a certain line segment (red in the picture) tangent to the circle of radius 1. The length of the brown line is $\cos \theta$ and that of the green line is $\sin \theta$, which by similar triangles gives you the equation \[\tan\theta=\frac{\sin\theta}{\cos\theta}\]

term

The word term is used is several ways in math English.

A constituent of a sum or sequence

A term is one constituent of a sum or sequence (finite or infinite), in contrast to a factor, which is a constituent of a product.

Example

The terms in the expression ${{x}^{2}}+2x+1$ are ${{x}^{2}}$, $2x$ and $1$. 

A symbolic expression denoting a mathematical object

This is discussed here.  Note that this use and the preceding one conflict. For example, the expression "${{x}^{3}}\left( {{y}^{2}}-1 \right)$" contains symbolic terms ${{x}^{3}}$and ${{y}^{2}}-1$, which would in math English be called factors rather than terms. 

In terms of

"Writing a function in terms of $x$" means giving a defining equation containing $x$ as the only variable (but it may contain parameters).

the

The English word "the" is called the definite article. It is used in forming definite descriptions: phrases that describe a particular individual.

• The particular individual may be known from context.
• If you hear someone say "I saw the boy we talked to the other day" the speaker expects the listener to know who the speaker means. 
• If they say "I saw a boy with a straw hat", the speaker does not expect the listener to know which boy they mean ("a boy" is an indefinite description).
• Definite descriptions can also be formed with "this" and "that". 

The definite article and setbuilder notation

A set $\left\{ x|P(x) \right\}$ is often described in setbuilder notation with a phrase such as "the set of $x$ such that $P(x)$". This means the set is the set of all $x$ for which $P(x)$ is true.

Examples

• The set described by the phrase "the set of even integers" is the set of all even integers.  Beginners in abstract math classes sometimes miss this.
• To the question, "Let $E$ be the set of even integers. Show that the sum of any two elements of $E$ is even", students have given answers such as this: "Let $E= \{2,4,6\}$. Then $2+4= 6$, $2 + 6 = 8$ and $4 + 6 =10$, and $6$, $8$ and $10$ are all even."
• That answer is wrong, because $E$ is not "the set of all even integers". It would be better to word the question as, "Let $E$ be the set of all even integers." But students need to know that the "all" is frequently omitted.

theorem

To call an statement a theorem is to claim that the statement has been proved.

• In texts the proof is usually given after the theorem has been stated. In that case (assuming the proof is correct) it is still true that in real time the theorem "has been proved"!  
• Some authors use the word "theorem" only for assertions they regard as important, and use the word proposition for less important ones. See also lemma and corollary.
• Theorems, along with mathematical definitions, may be set off in the text with a box or a different typeface.

theory of functions

In older mathematical writing, the phrase theory of functions refers by default to the theory of analytic functions of one complex variable.

tilde

transformation

trigonometric functions

Degrees and radians

If you write "$\sin x$" meaning the sine of $x$ degrees, you are not using the same function as when you write "$\sin x$", meaning the sine of $x$ radians. They have different derivatives, for example:  Let $\sin x$ denote the sine of $x$ radians and $\text{sind}\ x$ the sine of $x$ degrees, so that $\text{sind}\,x=\sin \frac{\pi }{180}x$.  Then $\frac{d\sin x}{dx}=\cos x$ but $\frac{d\,\text{sind}\,x}{dx}=\frac{\pi }{180}\cos x$.

The same remark may be made of the other trig functions.  In postcalculus pure mathematics "$\sin x$" nearly always refers to the sine of $x$ radians, often without explicitly noting the fact. This is not true for texts written by non-mathematicians, who at least often use the degree sign to tell you their meaning.

Variations in usage

• In the USA you calculate the sine function on the unit circle by starting at $(1, 0)$ and going counterclockwise: the sine is the projection on the y-axis.
• In some other countries (for example, Hong Kong secondary schools) you may start at $(0,1)$ and go clockwise (the sine is the projection on the x-axis).  I learned this from students. 
• Students educated in Europe may not have heard of the secant function. They would simply write ${{\cos }^{-1}}$.
• You usually write evaluation of trig functions by juxtaposition (with a small space for clarity), thus:  $\sin x$ instead of $\sin(x)$. Students sometimes read "$\sin x$" as multiplication.

See also logarithm and tangent.   

trivial

About theorems

A theorem is said to be trivial to prove or trivially true

• a) If the theorem follows by rewriting using definitions, or
• b) If the speaker’s mental representation of the mathematical object mentioned in the theorem makes the truth of the fact immediately perceivable.   

Example:

Here is a scenario that exemplifies (a):

• A textbook defines the image of a function $F:A\to B$ to be the set of all elements of $B$ of the form $F(a)$ for some $a\in A$.
• It then goes on to say that $F$ is surjective if for every element $b$ of $B$ there is an element $a\in A$ with the property that $F(a) = b$.
• It  then states a theorem, or give an exercise, that says that a function $F:A\to B$ is surjective if and only if the image of $F$ is $B$.
• The proof follows immediately by rewriting using definitions.
• The instructor calls the proof trivial and goes on to the next topic.
• Some students are totally baffled.

I have seen this happen several times with this and other theorems.  This sort of incident may be why many intelligent people feel they are "bad at math".

• When a class is first introduced to proof techniques the instructor should explicitly describe rewriting by definitions with several examples.
• After that, the instructor can say that a proof follows by rewriting by definitions and make it clear that the students will have to do the work (then or later).
• Such a proof is justly called "trival" but saying it is trivial is also a putdown if no one has pointed out the procedure of rewriting by definitions.

Example:

This example illustrates (b).

Theorem: Let $G$ be a finite group and $H$ a subgroup of index $2$ (meaning it has half the number of elements of the group).  Then $H$ is normal in $G$.

Basic facts about groups and subgroups learned in first semester abstract algebra:

• A subgroup of a group determines a partition consisting of left cosets and another partition of right cosets, each (in the finite case) with the same number of elements as the subgroup.
• A subgroup is a left coset of itself and also a right coset of itself.
• If every left coset is also a right coset and vice versa (so the two partitions just mentioned are the same), then by definition the subgroup is normal in the group.

Now if $H$ has index $2$ that means that each partition consists of two cosets. In both cases, one of them has to be $H$, so the other one has to be $G\setminus H$, which must therefore be a left and right coset of $H$. So $H$ is normal in $G$.

So once you understand the basics about cosets and normal subgroups, the fact that $H$ has to be normal if it is of index $2$ is "obvious". I don't think you should call this "trivial". Best to say it is "obvious if you have a clear understanding of cosets of groups".

About mathematical objects

A mathematical structure is said to be trivial if its underlying set is empty or a singleton set. In particular, a subset of a set is nontrivial if it is nonempty. I have not found an example where "nontrivial subset" means it is not a singleton.  See also proper.

Note: "Trivial" and "degenerate" overlap in meaning but are not interchangeable.  What is called "degenerate" seems to depend on the mathematical specialty.

truth value 

The truth value of a statement is "true" if the statement is true and "false" if it is false.

Example

The truth value of "$7 + 3 = 10$" is "true" and the truth value of "$7 + 3 = 11$" is "false".

• In computing science, "True" may be represented by $1$ and "False" by $0$.
• The truth value of a statement can be thought of as a function from the set of statements to the set $\{0,1\}$ (or to {"true","false"}).
• See also Boolean Algebra.

Assertions with variables in them need not be either true or false. 

Example

If $n$ is an integer, the statement "$n \gt 7$" is neither true nor false as long as nothing is known about $n$.  If you know $n = 42$, then you know that "$n \gt 7$" is true.  If you know that $n \gt 4$, you still don’t know the truth value of "$n \gt 7$".  In this situation, some authors say the truth value is indeterminate.

In formal mathematical logic, a statement or an assertion becomes a math object, although this point is not often made clear in logic texts. An assertion with variables is a variable math object whose values range over $\{\text{true},\text{false},\text{indeterminate}\}$. See my post on variables. Atish Bagchi and I have called traditional mathematical logic string-based logic in this paper (which proposes a graph-based logic).

two

In mathematical discourse, two mathematical objects can be one object!  This is because two distinct names for a particular kind of math object usually are allowed to have the same value unless some word such as "distinct" is used to ensure that they are different.

Examples

• "The sum of any two even integers is even".  In this statement, the two integers are allowed to be the same.   It is true in any case. 
• "Are there two positive integers $m$ and $n$, both greater than $1$, satisfying $mn=9$?". The only answer is $m=n=3$. This question should never be stated that way. Either say "two distinct positive integers" or don't ask the question (in my opinion).

Once upon a time (before October 2006), Wikipedia’s definition of boolean algebra said until recently that a boolean algebra has "…two elements called $0$ and $1$…"  It did not say they must be distinct in the axiom but in the examples it said that the two-element boolean algebra is the simplest example.

There is more detail, including citations, in my post "Two".

type

under

Under is used to name the function or relation just referred to in the sentence.  The reference may be indirect or implicit.

Examples

• "The set $\mathbb{Z}$ of integers is a group under addition."  
• "If $x$ is related to $y$ under the relation $E$, we write $x E y$."
• "Let $F$ and $G$ be functions defined on the real numbers.  If the value of $x$ under $F$ is greater than the value of $x$ under $G$ for every $x$, then we say that $ F \gt G$."   This can be translated as:  "If $F(x)\gt G(x)$ for every $x$, then we say that $ F \gt G$."

It is my impression that some mathematicians use "under" in this way a lot, but most mathematicians don’t use it at all.  This needs more lexicographical research than what I did for the Handbook article on "under"..

underlying set

A mathematical structure is likely to be defined as a set $S$ along with some other mathematical objects that obey certain axioms. The set $S$ is the underlying set of the structure.

• Groups and topological spaces are defined in this way.
• "Underlying" is used in analogous ways with other structures; for example, a ring has an underlying Abelian group.
• Underlying structures are a basic tool in category theory. They are in fact functors, called forgetful functors.

unique

• To say that an object satisfying certain conditions is unique means that there is only one object satisfying those conditions.   For example, there is a unique even prime, namely the integer 2.
• The Handbook, page 259, discusses the philosophical confusion connected with questions such as "Is there a unique set of integers".  Mathematicians normally talk as if there is a unique set, but when pressed by foundations questions may say things like "Well, there are many copies but let’s assume we have picked a particular one." 
• The word "unique" is misused by students; this is discussed here. See also up to.

unit, unit element, unity

• A unit in a ring may mean either the identity element of the ring, or an invertible element of the ring.
• However,"ring with unit" means ring with identity element.
• Unit element or unity most likely means an identity element. This requires lexicographical research.

unknown

One or more variables may occur in a constraint, and the intent of the discourse may be to determine the values of the variables that satisfy the constraint. In that case the variables may be referred to as unknowns.

Examples

• Find the values of $x$ for $3{{x}^{2}}-2x-5=0$.  Answer:  $x=-1\,,\,\,\,\,x=\frac{5}{3}$.
• Find the values of $x$ for which $3x+5\gt 2$.  Answer:  $x\gt -1$.

In both these problems $x$ would be called an unknown.

unpack, unwind

A typical definition in mathematics may make use of a number of previously defined concepts. To unpack or unwind such a definition is to replace the defined terms with explicit, spelled-out requirements.  See translation problem and rewrite using definitions.

Similarly a function may be defined by a complicated formula.  To unpack such a formula means investigating it piece by piece, or chunk by chunk. Zooming and Chunking has an example.

up to

Let $E$ be an equivalence relation. To say that a definition or description of a type of mathematical object determines the object up to $E$ (or modulo $E$) means that any two objects satisfying the description are equivalent with respect to $E$.

Examples

• An indefinite integral $\int{f(x)\,\,dx}$ is determined up to a constant. For example, $\int{2x\,\,dx}=x^2+C$, where $C$ is any real number. In this case the equivalence relation is that of differing by a constant.
• The statement "If $G$ is a finite group of order $n$ containing an element of order $n$, then $G$ is the cyclic group of order $n$" is a statement that defines $G$ up to isomorphism.   

• The object that is the result of evaluating a  function at an element $x$ of its domain is called the value, output or result of the function at $x$.
• The word value is also used to refer to the object denoted by a literal expression. Most commonly the word is used when the value is a number.

Example

The value of the function defined by $f(x)={{x}^{2}}+2$ is $11$. You could also say that the value of the expression $x^2+2$ is $11$.

vanish

A function $f$ vanishes at an input $a$ if $f(a)$ = 0. For example, the function $f(x):=x^3-27$ vanishes at $3$.  This (pretty common) usage can cause cognitive dissonance, since the function still exists!

variable

vector

The word vector has (at least) three different useful mental representations:

• An $n$-tuple.
• A quantity with length and direction.
• An element of a vector space.

Of course, the third representation includes the other two, but with some subtleties. For example, to think of an element of an abstract $n$-dimensional vector space as a $n$-tuple requires choosing a basis for the space. There is in general no canonical choice of basis.

In computer engineering, the word vector is often used to refer to an $n$-tuple of any sort of thing, not necessarily elements of a field (or numbers at all), so that the $n$-tuple may indeed not be a member of a vector space. I have heard this usage in conversation but could not find an unequivocal citation for it for the Handbook.

well-defined

Suppose a function $F$ is defined by an assertion "$F(X) = E$", where $X$ and $E$ are some expressions in the symbolic language.  $F$ is said to be well-defined if:

• WD1 $X$ denotes a specific element of the domain of $F$.
• WD2 Let $Y$ be a possibly different expression denoting the same element of the domain of $F$ that $X$ denotes.  Let $E’$ be the expression obtained from $E$ by substituting every occurrence of $X$ in $E$ by $Y$.  Then $E$  and $E'$ must denote the same mathematical object.

Note: Although we say "$F$ is well-defined", in fact "well-defined" is a property of the notation used in defining $F$, not of $F$ itself.  

Example

Let $F:\mathbb{R}\to\mathbb{R}$ be the function defined by $F(|x|)=x^2$. Then $F$ is well defined because $|x|=|y|$ if and only if $x=y$ or $x=-y$, so we can check that $F(|x|)=x^2$ and $F(|-x|)=(-x)^2=x^2$. Note that this does not work for $G(|x|)=x^3$ because then $G(|-1|)=(-1)^3=-1$ but $G(|1|)=1^3=1$.

when

When may mean "if"

Example

"When a function has a derivative, it is necessarily contin­uous."

Phrases containing "when"

These phrases

• when and only when
• exactly when
• precisely when

all mean if and only if.

Example

"A function has a derivative precisely when it is continuous" is false: for example, the absolute value function is continuous but doesn't have a derivative at $0$.

whenever

Whenever is also used to mean "if".  It may be used after another "if" to avoid having two "if"s in a row, for example

"A relation $\alpha $ is symmetric if whenever $x\alpha y$, then $y\alpha x$".

where

Where may be used in a special way in math English, sometimes without any connotation of location.

Examples

• "Definition: An element $a$ of a group is involutive if ${{a}^{2}}=e$, where $e$ is the identity element of the group."  Note that where carries no connotation of location in this sentence.  The statement "where $e$ is the identity element of the group" is a postcondition.
• "A point $x$ where $f'(x)=0.$" In my opinion, this usage also carries no connotation of location, but I suspect some may disagree.

without loss of generality

A proof of an assertion involving two elements $x$ and $y$ of some mathematical structure $S$ might appear to require consideration of two cases in which $x$ and $y$ are related in different ways to each other. For example for some assertion $P$, $P(x, y)$ or $P(y, x)$ could hold. However, if there is a symmetry of $S$ that interchanges $x$ and $y$ without affecting the property in question, you may need to consider only one case. In that case, the proof may begin with a remark such as,

"Without loss of generality, we may assume $P(x, y)$."

WLOG is an abbreviation for this phrase.

Example

Let $r$ and $s$ be real numbers. The mean (or average) of $r$ and $s$ is $\frac{r+s}{2}$. You would expect that the average is between $r$ and $s$ in value. Here is a formal proof that $\frac{r+s}{2}$ is between $r$ and $s$:

Assume without loss of generality that $r\lt s$. Then $\frac{r}{2}\lt\frac{s}{2}$, so \[r=\frac{r+r}{2}\lt\frac{r+s}{2}\lt\frac{s+s}{2}=s\]

The reason you can assert WLOG in this case is that if $r\gt s$ then you can simply switch their names. You can do this because if a number is between $r$ and $s$ then it is between $s$ and $r$. That property of betweenness is the symmetry referred to in the definition of WLOG above.

witness

A witness to an existential statement of the form $\exists x\,(P(x))$ is an object $w$ for which $P$($w$) is true.  For example, a witness to the statement "There is an integer $x$ for which ${{x}^{2}}\gt 5$" is the integer $3$ (among many others).  The integer $2$ is not a witness to this statement because ${{2}^{2}}=4$, which is not greater than $5$.  On the other hand, the statement "There is an integer $n$ for which $n\gt n$" is false, so it has no witnesses at all.

• The symbol $\mathbb{Z}$ usually denotes the set of all integers. Some authors use $\mathbb{I}$.

• A lowercase $z$ is commonly used for a complex variable.  The customary notation is that $z=x+iy$, where $x$ and $y$ are real variables.

Notes

• The letter Z is pronounced "zee" in the USA and "zed" in the United Kingdom and in much of the Commonwealth.

• The specification language Z is called "zed" even by many Americans.

zero

The number zero

The number zero is an integer. It is the number of elements in the empty set. In American college and university usage it is neither positive nor negative, but many college students show confusion about this.

There is more about zero in the abmath entry for the empty set symbol.

Zero of a function

A zero of a function $f(x)$ of one variable is an element $c$ the domain of $f$ for which $f(c) = 0$.  It is common to use the word root for such an element $c$. 

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