 abstractmath.org

GLOSSARY

### 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.

In old books sometimes the plural of lemma is lemmata.

### let

“Let” is used in several different ways in mathematical English.  In many cases, assume, suppose and if can be used instead of “let”.  See usage.

#### 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.

##### 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.

##### Remark

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.

##### Remarks

Define is sometimes used in this sense of “let”; more about that here.

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.

If, assume and suppose can be used in this situation, with requisite changes in syntax.

#### To consider successive cases

##### Example

“Let n > 0.... Now let If, assume and suppose seem to be more common that “let” in this use.  #### To introduce the precondition of a definition.

##### Example

“Definition:   Let  n be an integer. Then  n is even if  n is divisible by  2.”  If, assume and suppose can be used here.

#### 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 and .”    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   The following is a more explicit version of the same assertion:

“Let the noncommutative group G be given. Since G is noncommutative, the collection is nonempty. Hence we may choose a member (x,y) of this set... “  ##### Example

In proving a function is injective, you might begin with “Let be elements for which ”.  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 are witnesses is implied by the assumption that F is not injective.        ##### Remark

The choice of witness may be a parametrized choice.

##### Example

Assuming  c is a complex number:

“Let  d be an  nth root of c.”

#### “Let” in definitions

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

##### Example

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

##### Remark

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.

#### Usage

In many cases, assume, suppose and if can be used instead of “let”.  Let, assume, suppose and if use varying syntax:  “is” instead of “be” for “assume” and “suppose”, and the sentences must be combined into one sentence with “if”.

##### 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.

##### Other differences

The four words “assume”, “suppose”, “if” and “let” are used in subtly different ways in math.  I’d like to know if anyone has investigated this.

Atish Bagchi.

### linear

a)   A function is linear if it is of the form , 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 is linear (or an A-linear map) if for all a and a′  in A and m and m′  in M, .    Note that an F vector space over a field F is a module over F.  In particular, is a vector space over itself, so the two definitions of linear for functions conflict.  In particular, an example of (a) with b ≠ 0 is not an example of (b).  This 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.

Mariana Montiel.

### logarithm

For real numbers a and x with a positive and not equal to 1,  the expression denotes the number y for which .   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”.

Authors commonly don’t tell you

which basis for logarithms  they are using.