abstractmath.org

GLOSSARY

**¨ ****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.

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

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.

“An integer
divisible by 4 is divisible by

A proof could begin this way:

“Let *n* be an integer
divisible by

This introduces a new variable symbol *n * and constrains
it to be divisible by

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

Then the proof could begin

“Let *n* be divisible
by

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.

**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.

“Let *n *> 0....
Now let ” **If**, **assume**
and **suppose** seem to be more common
that “let” in this use.

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

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.

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 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.

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.

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... “

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.

The choice of witness may be a parametrized choice.

Assuming *c* is a complex
number:

“Let *d* be an *n*th root of* c*.”

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

“Let an integer be even if it is divisible by

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.

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”.

**All
these sentences have the same mathematical content.**

¨
“Let *S
be*
an odd integer. Then

¨
“Assume *S is* an odd integer.
Then

¨
“Suppose *S is* an odd integer.
Then

¨
“If *S* **i****s** 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.**

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.

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.

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. **