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) (A

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
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

Thus, (A

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

= 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
= {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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