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OTHER SYMBOLS USED IN MATH

Contents

Introduction

• This list includes only the major meanings of the most common symbols, according to my judgment.
• The list of mathematical symbols in Wikipedia includes many other symbols but does not include all, or even most of, the meanings of the symbols.
• Like many other parts of abstractmath.org, in depth lexicographical research is needed (anyone have ten million dollars?) See Remarks about usage.
• More information about mathematical symbols may be found in individual entries in Mathworld and in Wikipedia.

Accented characters

Mathematicians frequently use an accent to create a new variable from an old one. This is usually done to denote a mathematical object with some specific functional relationship with the old one.

• The most commonly used accents over the letter are arrow, bar, check, dot, hat and tilde.
• prime and star may be used after the letter as a superscript.
• slash may be used across a relational symbol.

All of the accents except arrow and slash (as far as I know) have more than one meaning. In the list below, I mention some common meanings, but mathematicians may and do define other meanings for the symbols.

Arrow

• Expressions such as $\vec{a}$, $\vec{m}$, $\vec{v}$, $\vec{x}$ usually denote vectors.
• A variant is $\overset{\scriptscriptstyle\rightharpoonup}{x}$. (The symbol above the $x$ is called a "harpoon").
• In print, vectors are usually printed in boldface, for example "$\textbf{x}$", but the arrow version is easier to use at the blackboard.
• In LaTeX, $\vec{x}$ is coded as "\vec{x}".
• The arrow is also used stand-alone in arrow notation.

Bar

• For a variable $x$, "$\bar{x}$" is pronounced "x bar".
• In probability and statistics, the bar may be used to denote the sample mean of a random variable.
• In many branches of math, the bar may be used to denote the closure (however it is defined) of a substructure of a mathematical structure.
• The bar is also used to denote the conjugate of a complex number.
• The typographical name of the symbol is macron.
• In LaTeX, $\bar{x}$ is coded as "\bar{x}".

Check

• The symbol "ˇ" over a letter is commonly pronounced "check" by mathematicians. For example, $\check{c}$ is pronounced "c check".
• The typographical name for the symbol is caron or haček.
• The haček is used in Čech cohomology and occasionally in other fields.
• In LaTeX, $\check{x}$ is coded as "\check{x}" in math mode.

Dot

• The expression $\dot{x}$ is pronounced "x dot".
• In math, by far the most common use of "$\dot{x}$" is to denote the derivative of the variable $x$ with respect to time.
• The dot is also used as a binary operator to denote multiplication, as in "$x\cdot y$", and to denote the dot product of vectors.
• In LaTeX, $\dot{x}$ is coded as "\dot{x}", and $x\cdot y$ is coded as "x\cdot y".

Hat

• The expression $\hat{x}$ is pronounced "x hat".
• The hat is used to mean a very large number of different things. One that seems to me very common is to indicate the Fourier transform of a function.
• A peculiar usage is to indicate that an item in a list is omitted. For example, \[(x_1,x_2,x_3,\hat{x}_4,x_5)\] means \[(x_1,x_2,x_3,x_5)\]
• The typographical name of the symbol is circumflex.
• In LaTeX, $\hat{x}$ is coded as "\hat{x}".

Prime

• The symbol "$'$" is pronounced "prime" (USA) or "dash" (outside USA). For example, $x'$ is pronounced "x prime" or "x dash". $x''$ is pronounced "x double prime".
• If $f$ is a differentiable function, $f'$ usually denotes its derivative.
• If $x$ is a variable, $x'$, $x''$ and so on may be defined as new variables of the same type. I have seen this usage bother beginning abstract math students, who expect it to mean the derivative.
• The example of a proof in Presentations of proofs shows the prime and double prime in use.
• See more about the prime symbol in MathWorld.

Slash

• A slash across a relational symbol means the resulting statement is false. For example, "$x\ne y$" means that the statement "$x = y$" is false; in other words, $x$ is not equal to $y$.
• In LaTeX, "$\ne$" is coded as "\ne".
• To get $x\!\!\!/$, type "x\!\!\!/". The slash comes out a little off center with some letters.

Star

The asterisk "*" may be used independently or as a modifier of a variable. In either case (except when it denotes multiplication) it is pronounced "star".

Star as modifier

• If $a$ is a variable, $a^\ast$ may denote the result of applying a specific involution to $a$, for example the complex conjugate or the adjoint.
• If $a$ is a character, $a^\ast$ may denote any number of repetitions of the character.
• For these uses, the LaTeX code for $a^\ast$ is "a^\ast". Note that LaTeX also allows $a^\star$, which is typeset by "a^\star", but I don't know of examples of its use in math.

Used independently

• A "*-algebra" is an algebra with a designated involution called "*".

• In programming languages "*" is used to denote multiplication. More here.

• The word "star" may be used as a mathematical adjective without being associated with asterisks, as for example with star polyhedra.

Tilde

• The symbol "~" is pronounced "tilde" (till-day or till-dee), or informally "twiddle" or "squiggle".
• The tilde may be placed on top of a variable, for example "$\tilde{x}$" with several possible meanings.
• Before a number the tilde is used to mean "approximately". "~42" means "approximately 42".
• $\sim \,,\,\,\approx \,,\,\,\simeq \,,\,\,\cong$ are all used to denote binary relations. More about this below.
• Other uses are discussed in MathWorld.
• In LaTeX, you code $\tilde{x}$ as "\tilde{x}" and $x\sim y$ as "x\sim y".

Quantifiers

Quantifier symbols are used in logic.

• $\forall $ is the universal quantifier. Discussed in the article Universally True.
• $\exists $ is the existential quantifier. Discussed in the article Existential statements.

Usage

• Many mathematicians use the symbols "$\forall$" and "$\exists$" at the blackboard as abbreviations.
• Except in books and papers on logic, it is not common to see them in printed texts.

Pronunciation

Pronouncing expressions using the two quantifiers involves a difference in English syntax.

• $\forall n\,(n\text{ is even)}$ is pronounced "For all $n$, $n$ is even."
• $\exists n\,(n\text{ is even)}$ is pronounced "There is an $n$ such that $n$ is even."
• In LaTeX you code "$\forall$" as "\forall" and "$\exists$" as "\exists".
• See also the articles on notation for connectives and negation.

• "$:=$" has a unique meaning.
• "$=$" has a unique meaning in math, but not in computing.
• Each of the others have at least two common meanings.
• There are many other symbols denoting some form of equivalence as well. You can find them in Wikipedia's list of mathematical symbols.

$\boldsymbol{=}$

• The equals sign "=" has a standard meaning in math: "x = y" means x and y are two different names for the same mathematical object.
• In some programming languages, "$=$" is an assignment operator. In such a language, the statement "$x=42$" means that the variable $x$ now has the value $42$ (the same as saying "Let $x=42$"). Many such languages use "$==$" for the mathematical meaning of the equals sign.
• An expression containing an equals sign may have various intents, and determining those intents is something students must learn how to do.
• For more information, see equation in the Glossary, equation, equal sign, and equality in Wikipedia, and the article When is one thing equal to some other thing? by Barry Mazur.

$\boldsymbol{:=}$

• This symbol is called colon equals.
• It means means "is defined to be". For example, "$f(x):={{x}^{2}}+1$" means that $f(x)$ is defined to be ${{x}^{2}}+1$ and "$A:=\{1\,,\,3\,,\,5\,,\,6\}$" means that $A$ is defined to be the set $\{1,3,5,6\}$.
• The scope of definitions given this way are usually small, for example the current paragraph or section.
• This usage originated in computing science and now many mathematicians use it.
• Two other symbols are also used to mean "is defined as", illustrated by "$x\equiv42$" and "$x{\overset{\text{def}}{=}}42$".

$\boldsymbol{\equiv}$

• The symbol "$\equiv$" most commonly denotes a modular congruence or a geometric congruence. Note that the symbol "$\cong$" is also commonly used for these meanings.
• The Wikipedia article Triple bar describes other less common meanings of this symbol.
• I have never heard a mathematician use the name "Triple bar".
• In LaTeX, "$\equiv$" is coded as "\equiv".

$\boldsymbol{\sim}$

• "$\sim$" used between two math expressions is called the tilde operator.
• For functions $f$ and $g$, the statement "$f\sim g$" means that $f$ is asymptotic to $g$.
• "$\sim$" is one of many symbols, listed in the Wikipedia article on approximation, used to indicate that one number is approximately equal to another. Note that "approximately equal" is reflexive and symmetric but not transitive.
• "$\sim$" is one of many symbols used in logic to indicate negation.

$\boldsymbol{\approx}$

• For numbers $x$ and $y$, the statement "$x \approx y$" means that $x$ is an approximation to $y$.
• For functions $f$ and $g$, the statement "$f \approx g$" means that $f$ and $g$ are asymptotic.
• In LaTeX, "$\approx$" is encoded as "\approx".

$\boldsymbol{\cong}$

• The symbol "$\cong$" may denote that two math objects are isomorphic, a modular congruence or a geometric congruence. These are all equivalence relations.
• The symbol may be used with other meanings, including "approximately equal to".
• In LaTeX, "$\cong$" is coded as "\cong".

Inclusions

The symbols $\subset \,,\supset \,,\subseteq \,,\supseteq \,,\varsubsetneq \text{, and}\varsupsetneq $, along with some less common symbols, all denote relations concerned with inclusion.

$\boldsymbol{\subseteq}$

• "$A\subseteq B$" means that every element of $A$ is an element of $B$. This can be worded as:
• $A$ is included in $B$, or
• $A$ is a subset of $B$.
• The LaTeX code for "$\subseteq$" is "\subseteq".

"$A\subseteq B$" specifically includes the possibility that $A = B$. Note that that is the standard meaning of "subset": "$A$ is a subset of $B$" always allows $A=B$ and always allows $A$ to be the empty set.

$\boldsymbol{\subset}$

• The statement "$A\subset B$" has two meanings in math writing and, at least in the research literature, authors are not likely tell you which one they are using.
• "$A\subset B$" may mean exactly the same thing as "$A\subseteq B$", including the possibility that $A = B$. This meaning is very common in research papers and it occurs sometimes in textbooks too.
• "$A\subset B$" may mean $A\subseteq B$ but $A\ne B$, in analogy with "$\le$" and "$\lt$". This meaning is common in textbooks and seems to be universal in high school math texts.

When you see "$A\subset B$",
make sure you know what the author means.

• The LaTeX code for "$\subset$" is "\subset".

$\boldsymbol{\supset}$

• "$B\supset A$" means $A\subset B$ in either of its meanings.
• It is pronounced "$B$ includes $A$" or "$B$ is a superset of $A$".
• When $A$ and $B$ are statements (not sets) $B\supset A$ may mean "If $B$, then $A$". See the abmath article on Conditional assertions.
• The LaTeX code for "$\supset$" is "\supset".

$\boldsymbol{\supseteq}$

• $B\supseteq A$ means $A\subseteq B$.
• This can be phrased as "$B$ includes $A$" or "$B$ is a superset of $A$".
• The LaTeX code for "$\supseteq$" is "\supseteq".

$\boldsymbol{\varsubsetneq}$

• "$A\varsubsetneq B$" means that every element of $A$ is an element of $B$ and there is an element of $B$ that is not an element of $A$.
• This symbol thus means the same thing as one of the two meanings of "$\subset$", and in my opinion should be more widely used.
• The LaTeX code for "$\varsubsetneq$" is "\varsubsetneq".

Miscellaneous symbols

$\boldsymbol{\times}$

• "$a\times b$" can mean the product of numbers $a$ and $b$, the product of matrices, the cartesian or categorical product of mathematical objects, or the vector product of 3-dimensional vectors. There is more information about this symbol in the article overloaded notation.
• The product of numbers or matrices can also be represented by juxtaposition ("$ab$" instead of "$a\times b$") or (in Mathematica) by a space, but that is not true of the cartesian product or the vector product.
• The LaTeX code for "$\times$" is "\times".

• The phrase "$x\in S$" means $x$ is an element of the set $S$.
• Some mathematicians write this symbol as $\varepsilon $ (one form of the Greek letter epsilon.) Sometimes they even call it "epsilon", but it is really intended to be a different symbol.
• The LaTeX code for "$\in$" is "\in".

$\boldsymbol{\varnothing}$

• The symbol "$\varnothing$" denoted the empty set.
• Another symbol used for the empty set is "$\emptyset$".
• Computer people often use "$\emptyset$" to denote zero, in order to distinguish it from "$0$". This leads some computing science students to think that the empty set is zero (I speak from repeated personal experience.)
• Some approaches to the foundations of mathematics define the number $0$ to be the empty set, but it is entirely wrong to think of $0$ and the empty set as the same. That is a violation of type. The empty set is a particular set and $0$ is a particular number. See the Wikipedia articles on type theory and on type systems.
• The symbol "$\varnothing $" is not the Greek letter phi, written $\phi $. (You will occasionally hear mathematicians refer to the empty set as "phi", which is historically incorrect).
• The LaTeX code for "$\varnothing$" is "\varnothing".

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