abstractmath.org

GLOSSARY

The word **tangent** may refer
either

¨ to a straight
line in a certain relation to a differentiable curve, or

¨ to a certain trigonometric
function.

These two meanings are related:
the trigonometric tangent of an angle (SOC in the picture) is the length of a
certain line segment (TD in the picture) tangent to the circle of radius 1.

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

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**.

The terms in the expression are ,
2*x** *and 1.

¨
This is discussed here.
**Note that this use and the
preceding one conflict. ****For
example, ****the expression “ **** ” contains symbolic terms **** and ****, which
would in math English be called ****factors**** ****rather
than terms.**

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

The expression *implicitly* determines *y* in terms of *x*. *Explicitly*, y is given **in terms of ** *x *by
*.*

Abbreviation for "the following are equivalent".

The phrase "**that is**" may be used to indicate
that **what follows
is equivalent
to what precedes**, usually when the equivalence is essentially a
rewording.

¨
"We have shown that *xy < *0, that is, that *x *and
*y *are nonzero and of opposite sign."

¨ "Then *n *= 2*k*, that is, *n *is even."

"That is" is sometimes used
in the situation that what after is an **explanation of the
significance of what comes before. **The
explanation may be a rewording as above, so that these two
usages overlap.

"If the function is a polynomial, that is, easy to calculate,
numerical estimates are feasible."

The |

The word "**the**" is
called the **definite article**. It is
used in forming definite descriptions:
phrases that describe a particular individual. The particular individual is often 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
what the speaker means. If they say “I
saw a boy with a straw hat”, the speaker does **not ****expect
the listener to know what they mean (“a boy” is an**** indefinite
description****). Definite descriptions can also be formed with
“this” and “that”. **

In math English, both the indefinite article and the definite article can have the force of universal quantification. Examples are given under universal quantifier.

A set is often described in setbuilder notation with a phrase such as
"the set of x such that P(x)". In particular, this set is the
set of **all **
x for which P(x) is true..

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

This answer is wrong, because **E is not
the set of all even integers**.

The definiendum of a definition may be a definite description.

The word “**the**” can make you believe that you should
know what it refers to. More about this
here.

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 refer only to
assertions they regard as important as theorems, 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.

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

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
sind *x *the sine of *x *degrees, so that . Then but .

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.

¨ In
the **counterclockwise**: the sine is the projection on the y-axis. In some other countries (for example, **clockwise **(the sine is the projection on the x-axis). I learned this from students.

¨
Students educated in

¨ 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
this as multiplication.

See also logarithm
and tangent.

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.

¨ I prefer “obvious” for meaning (b). “Trivial” is unnecessarily put-downy.

¨ These definitions are based on my own observation and are not justified by lexicographical citations.

Here is a scenario that exemplifies (a):

¨
A textbook defines the **image** of a
function to be the set of all elements of* B* of the form *F(a)* for some .

¨
It then goes on to say that *F* is **surjective **if for every element *b* of *B* there is an
element with the property that *F*(*a*) = *b*.

¨
It then
states a theorem, or give an exercise, that says that a function is surjective if and only if the image of* F* is *B.*

¨
**The proof
follows immediately by rewriting using definitions***. *

**¨
****Some students are totally baffled.**

¨
**The
instructor calls the proof trivial and goes on to the next topic.**

If you are bad at math, don’t blame your brain, blame your teachers. --Paul Cox |

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”.

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.*

If part of your basic picture of a group and a subgroup is
that the subgroup determines a partition consisting of left cosets and another partition of right cosets,
each with the same number of elements as the subgroup, then *H* having index 2 immediately means that
every left coset is a right coset and vice versa, which is equivalent to *H *being normal. This exemplifies (b).

A function may be
called **trivial** if it is the identity
function or a constant function, and possibly
in other circumstances.

A **solution
to an equation** may be said to be **trivial**
if it is 0 or 1. (This depends on the
circumstances. If someone showed that
the cosmological constant
is 0 that would **not ****be called trivial.)** There may be other
situations in which a solution is called
"trivial" as well.

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.

"Trivial" and “degenerate”
overlap in meaning but are not interchangeable. What is called “degenerate” semes to depend on
the mathematical specialty.

The **truth value** of
a statement is “true” if it is true and “false” if it is false. For example, the truth value of “7 + 3 = 10”
is “true” and the truth value of “7 + 3 = 11” is “false”. Observe that this is in fact a function from
the set
of statements to the set {“true”, “false”}.

**Assertions
with variables in them need not be either true or false**. For example, if *n* is an integer, the statement “*n
> 7*” is neither true nor false as long as nothing is known about *n*.
If you know *n = *42, then you
know that “*n > 7*” is true. If you know that *n > 4*, you still don’t know the truth value of “*n > 7*”. In this situation, some authors say the
truth value is **indeterminate.**

In mathematical discourse, two mathematical
objects can be one object! This
is because two identifiers used for them **usually are allowed to have the same value**
unless some word such as “distinct” is used to
ensure that they are different.

**¨
**"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.
**

**¨
**“The
difference of any two distinct even integers is a nonzero even integer.” This statement is true but it requires the
word “distinct”. Many mathematicians,
including me, would regard the statement with “distinct” omitted as **either ****false****
or ****badly
written****.**

¨ Another example occurs under transitive.

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. (It says “distinct” now 18 October 2006.)