abstractmath.org
GLOSSARY
Posted 2 January 2009
I
¨ I (bold) may denote the unit interval, the set of
real numbers x for which 
. It may however denote any bounded interval of
real numbers.
¨ I may be used as the name of an arbitrary
index set.
¨ For some authors, I (bold) or (blackboard
bold) means the set of integers;
however, Z or is also used for this.
¨ I may denote the identity function. “id” is also used for this.
¨ I may denote the identity matrix for a
dimension given by context.
¨
i (usually lowercase) denotes one
of the square roots of 1. (The other one is then i).
In engineering, j is very
frequently used instead of i for
this.
These various meanings are quite commonly used without explanation.
The expression “i.e.” means that
is. This abbreviation is confused by
many readers and writers (including writers in the research literature) with
“e.g.”, which means “for example”. This is a very significant
difference in meaning. Consider
¨ “Let n be an even integer, e.g. 6.” Good
¨
“Let
n be an even integer, i.e. 6.” Bad. It
suggests that 6 is the only
even integer.
¨ “Let
n be an even integer, e.g. divisible
by 2.” Bad. It
says, “Let n be an even integer, for example
divisible by 2.” The reader
expects the phrase after the comma to provide a particular integer that is
even, but it doesn’t.
¨
“Let n be
an even integer, i.e. divisible by 2.” Good
If you desperately want to be
understood, you should never use “i.e.” and “e.g.”
Too many readers don’t know
what they mean.
To identify an object A with another
object B is to regard them as the same object. In
particular each point in A is
regarded as the same as a particular point in B. In most cases, the author
has in mind a particular way of matching the points in A and the points in B. This “particular way” may be presented
informally or as a formal mathematical construction.
Examples
¨
You may regard each point of the real plane
as having specific
coordinates, an ordered pair of real numbers. (“Consider the point with coordinates (2.3,
-1.45).”) On the other hand, you may
think of the points of the real plane as identified with ordered pairs
of real numbers. (“Consider the point
(2.3, -1.45).”) These are two ways of thinking about the
plane (an example of conceptual blending). Both Wikipedia and MathWorld define the
plane in such a way that its points are represented by pairs of real
numbers. That is desirable in a
definition since it looks forward to the generalization to manifolds.
But you can think about
the plane either way and get away with it.
¨ The Möbius strip may be constructed by identifying
or gluing the edge
of the unit square with the edge
in such a way that (x, 0) is
identified with (1 x,
1). This can be thought of as
saying that the coordinates (x, 0) and (1 x,
1) refer to
the same point. This
is an example of identifying objects by a formal mathematical construction.
Usage
In ordinary English, “identify” means to give a name to. This presumably could cause cognitive
dissonance.
This word has three common meanings.
Equation that always holds
An identity in this sense is an equation that holds between two
expressions for any
valid values of the variables in the expressions. Thus, for real
numbers (in fact for complex numbers), the equation
is an identity.
Fine point
Consider the assertion “If f(x) is a constant
function, then for any x, ”. The equation is conditional
on the hypothesis “f(x) is a constant function” being true. In this situation, the equation would not be
called an identity. This requires
further lexicographical investigation.
Usage
Sometimes in the case of an identity the
symbol ≡ is used instead of the equals sign.
Identity element of an algebraic structure
If for all elements x in an algebraic
structure with binary operation ,
then e is an identity or identity element for the
structure. Such an element is also called a unit element or unity.
A ring with identity may mean a ring with identity element, but it can
also mean a ring subject to an algebraic identity.
Identity function
For a given set S, the function from S to S
that takes every element of S to itself is called the identity
function. (More about that here.) This is
an example of a polymorphic definition.
The identity function on S
is the identity element of the algebraic structure consisting of all
functions from a set S to itself with
composition as
operation.
In conditional assertions
The ways if is
used in stating conditional assertions is discussed here.
See also let, which covers assume and suppose as well.
In definitions
It
is a convention that the word if used
to introduce the definiens in a
definition means “if and only if”.
Examples
a: “An integer is even
if it is divisible by 2.”
Some authors regularly use if and only if or “iff” in definition, so the
definition above would be given as
b: “An integer is even
if and only if it is divisible by 2.”
But note:
The following statement is not a
definition, but an equivalence:
c: “An integer is divisible by 2 if and only if it is
divisible by 2.”
Even
though “if and only if” is used in both sentences, they don’t have the same status. In statement b the left side and the right side are not symmetric. The point of statement b is to give a meaning to the left side. Statement c is symmetric;
you could just as correctly say
“An integer is divisible by 2 if and only if it is divisible by 2.”
In the precondition of a definition
If can be used in the precondition of a definition to
introduce some requirements on an object mentioned
in the definition. For example, definition a above could have been worded this way:
“If n is an
integer, then it is even if it is divisible by 2.”
If P and Q are two assertions, the
statement “P if and only if Q” is an
assertion that P and Q
are equivalent.
This statement may be abbreviated “P iff
Q”.
image
See the article in Wikipedia.
imply
If P and Q are assertions, “P implies
Q” means exactly the same thing
as “If P, then Q”. This is discussed in
detail under conditional assertions.
In is used in mathematical discourse in all its English meanings, as well
as in some meanings that are peculiar to mathematics.
¨ “A is in B” can mean
¨ “A is in B” can mean .
¨ You may say “A is in B” when A is an equation whose solution set is included in B, or a geometric figure whose points
are included in B. Example:
"The unit
circle is in the Euclidean plane."
¨ You may say “A is in B” when A is a
subexpression of B. For example, x is a variable in ,
and is a term in .
¨ A is P
in B, where P is a property, may mean that A has property P with respect to B,
where B is a constituent of A or a related structure (for example a
containing structure). Thus in group theory you can say A
is normal in B, where A is a subgroup of the group B. As another example, is differentiable in y (and x).
¨ You may describe an
intersection using "in". For example, the sets {1, 2, 3, 4} and {1, 3, 5, 7} intersect in {1,
3} (or intersect in 1
and 3.)
in general
¨
Statements such as “In general, P” or “P is true in general” mean that P is true no matter what values you substitute for its variables.
¨
Statements such as “In general, not P” mean that P is not
necessarily true.”
Examples
¨
“The equation is true in general” means that for all x, is true.
¨
“In general,
not every subgroup of a group is normal in the group” means that an arbitrary subgroup of a group need not be normal in the group. The statement specifically implies that there are examples where the subgroup is not normal in the group.
Remark
In most circumstances, you can substitute generally for “in general”.
Let be a function.
¨
f is increasing
if for all x and y, if x < y then f(x)
< f(y).
¨
f is
decreasing if for all x and y, if x < y then f(x) > f(y).
¨
f is nonincreasing if for all x and y, if x < y then
.
¨
f is nondecreasing if for all x and y, if x < y then
.
These words can be used for functions from and to
other ordered sets, as well.
They are also used for sequences, since a sequence
is a function on its index set.
For example, a sequence is increasing if for all i and j in its index set, .
An inequality is
an assertion of the form ,
where s and t are terms and α is one of the relations <,
>, or .
An assertion of the form is not called an inequality.
inhabit
The phrase “A inhabits
B” can mean any of several things:
¨
A is an element of B.
¨
A is an entry in the list B.
¨
A is an expression in the delimiters B. (For example, “The function inhabiting the
curly braces is increasing.”)
Lives
in is used similarly in conversation, but I have found only a few citations
in print.
Functions can have a property called injective,
and so can modules. These two uses are unrelated.
This word has three different meanings.
Being an integer
Integral is used as an adjective to require that the
noun phrase it modifies denote an integer.
For example, 8 is an integral power of
2 but 9 is not. This is also used for
the more general notion of algebraic integer (MW, Wi).
Antiderivative
An integral of
a function is an
antiderivative of the function. It may
be called the indefinite integral to distinguish it from the definite
integral below. The indefinite integral
of f may be denoted by which
determines the integral only up
to a constant.
You may also refer
to a specific antiderivative using the form .
The word integral
is also used to denote a solution of a more general differential equation.
Definite integral
Integral is
also used to denote a definite integral: this operator takes an integrable function
and an interval (or more general space)
on which the function is defined and produces a number. The definite integral
of f on an interval [a, b] is denoted or .
Mathematicians may use the word intuitively in referring to some image
or metaphor they have about a type of object.
Examples
“Intuitively, the squaring function is a machine that turns,
for example, 3 into 9.”
They may admit
the intuition but point out its weaknesses:
“Intuitively, an open set in the plane is every point in a
region not including its boundary. In
fact, not all open sets fit this description, for example…”
is
See be.