abstractmath.org 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
tangent
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.
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 are
,
2x and 1.
A symbolic expression denoting a mathematical object
This is discussed here. Note that this use and the preceding one conflict.
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).
Example
The expression implicitly
determines y in terms of x.
Explicitly, y is given as a function of x by
.
TFAE
Abbreviation for "the following are equivalent".
that is
Indicating equivalence
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.
Examples
¨ "We have shown that xy < 0, that is, that x and y are nonzero and of opposite sign."
¨ "Then n = 2k, that is, n is even."
Indicating significance
"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.
Example
"If the function is a polynomial, that is, easy to calculate,
numerical estimates are feasible."
the
|
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
some mathematical discourse because the articles are used in English in some
fairly subtle ways. This is explained
in a handout from |
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”.
The definite article as universal quantifier
In math English, both the indefinite article and the definite article can have the force of universal quantification. Examples are given under universal quantifier.
The definite article and setbuilder notation
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..
Example
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.
Definite article in definitions
The definiendum of a definition may be a definite description.
Example
The word “the” can make you believe that you should
know what it refers to. More about this
here.
then
theorem
To call an assertion a theorem is to claim that the assertion has been proved.
Remark
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.
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.
thus
Thus can
mean that what follows is an example
of what precedes, or perhaps a consequence of
what precedes. This is pretty standard
academic English. This needs to be checked.
Also give examples.
tilde
The symbol " ~ " is an accent,
pronounced "tilde" (till-day or till-duh), or informally
"twiddle" or "squiggle".
¨ It may be used over a letter to create a new variable., for
example " ".
This would be pronounced "A tilde", "A twiddle" or "A squiggle".
¨
are all used to denote a binary relation.
More about this in Other
symbols.
¨Before a number the tilde is used to mean “approximately”. “~42” means “approximately 42”.
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. Spell this out. 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.
Variations in usage
¨ In
the
¨
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. Give link for
log.
trivial
About propositions
A theorem is said to be trivial to prove or trivially true
¨ If the theorem follows by rewriting using definitions, or
¨ If the common mental representation of the mathematical object mentioned in the theorem makes the truth of the fact immediately perceivable. (I would tend to use obvious for this possibility.)
Example
¨
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.
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”. (Here is a strong essay on the subject that I mostly agree
with.)
About mathematical objects
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 or 1 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. See proper.
Remark
"Trivial" and “degenerate”
overlap in meaning but are not interchangeable. What is called “degenerate” semes to depend on
the mathematical specialty.
truth value
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.
two
In mathematical discourse, two mathematical
objects can be one object! This
is because two identifiers used for them can usually 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.
¨ “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 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.)