1.1. Sets and Functions 5
Theorem 1.1.5. If A and B are subsets of a set X and Ac and Bc are their
complements in X, then
(a) (A
B)c
=
Ac

Bc;
and
(b) (A
B)c
=
Ac

Bc.
Proof. We prove (a) first. To show that two sets are equal, we must show that
they have the same elements. An element of X belongs to (A
B)c
if and only if
it is not in A B. This is true if and only if it is not in A and it is not in B. By
definition this is true if and only if x
Ac

Bc.
Thus, (A
B)c
and
Ac

Bc
have
the same elements and, hence, are the same set.
If we apply part (a) with A and B replaced by Ac and Bc and use the fact that
(Ac)c = A and (Bc)c = B, the result is
(Ac

Bc)c
= A B.
Part (b) then follows if we take the complement of both sides of this identity.
A statement analogous to Theorem 1.1.5 is true for unions and intersections of
collections of sets (Exercise 1.1.7).
Two sets A and B are said to be disjoint if A B = ∅. That is, they are
disjoint if they have no elements in common. A collection A of sets is called a
pairwise disjoint collection if A B = for each pair A, B of distinct sets in A.
Functions. A function f from a set A to a set B is a rule which assigns to each
element x A exactly one element f(x) B. The element f(x) is called the image
of x under f or the value of f at x. We will write
f : A B
to indicate that f is a function from A to B. The set A is called the domain of f.
If E is any subset of A, then we write
f(E) = {f(x) : x E}
and call f(E) the image of E under f.
We don’t assume that every element of B is the image of some element of A.
The set of elements of B which are images of elements of A is f(A) and is called
the range of f. If every element of B is the image of some element of A (so that
the range of f is B), then we say that f is onto.
A function f : A B is said to be one-to-one if, whenever x, y A and x = y,
then f(x) = f(y) that is, if f takes distinct points to distinct points.
If g : A B and f : B C are functions, then there is a function f◦g : A C,
called the composition of f and g, defined by
f g(x) = f(g(x)).
Since g(x) B and the domain of f is B, this definition makes sense.
If f : A B is a function and E B, then the inverse image of E under f is
the set
f
−1(E)
= {x A : f(x) E}.
That is, f −1(E) is the set of all elements of A whose images under f belong to E.
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