abstractmath.org
GLOSSARY
Posted 27 December 2008
Suppose a function F is defined
by an assertion “F(X) = E”,
where X and E are some expressions in the symbolic language. F is said to be well-defined if:
WD-1.
X denotes a specific
element of the domain of F.
WD-2.
Let Y
be an expression denoting the same element of the domain of F that X denotes. Let E’ be the expression obtained from E by substituting every occurrence of X in
E by Y. Then E and 
denote the same
mathematical object,
Note!
Although we say “F is well-defined”, in fact “well-defined” is a property of the notation used in defining F, not of F itself.
Example
For any integer n, let [n] denote the set of all integers of the
same parity as n (in other words, the
set of all even integers if n is even
and the set of all odd integers if n
is odd.) For example [0] = [4] = [42]
and [17] = [5]. Let denote the set {[0], [1]} . (This is fairly standard notation. See congruence.) Now define by
Then F is well-defined because an integer is even if and
only if its square is even. Here, X is the
expression “[n]” and E is the expression “ ”. In fact, F is the
identity function.
Example
Using the same notation as in the preceding example, if we defined G(n) = [the number of prime divisors of n], then G is not well-defined by WD-2, since we
would have [2] = [6] but G[2] = [1]
and G[6] = [2] = [0], and . Note that WD-1 does hold, since the number of prime divisors of an integer
is an integer and the expression “[n]” is defined
for every integer n.
Example
Let denote the set
of all nonempty subsets of the set of nonnegative integers. Thus and but .
Define by F(S) = the
smallest element of S. Thus and .
F
is well defined: WD-1 holds
because there is
a smallest
element of S and it is an integer, and WD-2 holds because no matter how you write a set
its smallest element is the same. For example,
If we have left out the requirement that consist
of the nonempty subsets of the set of nonnegative integers, then F would fail
WD-1 because there is no smallest element of the empty set. Similarly if you set to be the set of
all nonempty subsets of , because for example the set of all odd integers has
no smallest element.
Here is another example.
In math English, when may mean “if”.
Example
“When a function has a derivative, it is necessarily continuous.”
Usage
These phrases
¨
when and only when
¨
exactly when
¨
precisely when
all mean if and only if.
whenever
Whenever is
also used to mean “if”. It may be used
after another “if” to avoid having two in a row, for example
“A relation is symmetric if whenever then ”.
Where may be used in a special way in math English,
sometimes without any connotation of location.
Examples
¨
“Definition: An element a of a
group is involutive if ,
where e is the identity element of
the group.” Note that where
carries no
connotation of location in this sentence. The statement “where e is the identity
element of the group” is a postcondition.
¨
“A point x where f'(x) = 0
is a critical point.” In contrast to the previous usage, this usage carries a
connotation of location.
A proof of an assertion
involving two elements x and y of some mathematical
structure S might appear to require
consideration of two cases in which x and y are related in
different ways to each other; for
example for some assertion P, P(x, y) or P(y, x) could hold. However, if there is a symmetry of S
that interchanges x and y, you may need to consider
only one case. In that case, the proof may begin with a remark such as,
“Without loss of
generality, we may assume P(x, y).”
WLOG is an abbreviation for this phrase.
Example
“Theorem: Suppose m and
n are two distinct elements of the closed
interval of real numbers [a,
b].
Then the closed interval determined by m and n is shorter than [a, b]. Proof:
Without loss of generality,
assume m < n. The length of [a, b] is b a and the length of [m, n] is n m. Since
m and n are in [a, b], we now know that a < m < n < b. Hence n m < n
a < b a, as required.”
Of course, this theorem is obvious if you draw a
picture. But this proof technique is
useful in much more complicated circumstances.
Don’t
be insulted by trivial examples.
They
confirm that you understand something.
A witness to an existential statement
of the form is an object w for which P(w) is true. For example, a witness to the statement
“There is an integer
x for which ” is the integer 3 (among many others). The integer 2 is not a witness to this
statement because ,
which is not greater than 5. On the
other hand, the statement “There is an integer n for which ” is false, so it has no witnesses at all.