2 CHAPTER 1. SOME PREREQUISITE TOPICS

not so, there would have to exist a set A, such that /0 has something in it which is not in A.However, /0 has nothing in it and so the least intellectual discomfort is achieved by saying/0⊆ A.

If A and B are two sets, A\B denotes the set of things which are in A but not in B. Thus

A\B≡ {x ∈ A : x /∈ B} .

Set notation is used whenever convenient.To illustrate the use of this notation relative to intervals consider three examples of

inequalities. Their solutions will be written in the notation just described.

Example 1.1.1 Solve the inequality 2x+4≤ x−8

x≤−12 is the answer. This is written in terms of an interval as (−∞,−12].

Example 1.1.2 Solve the inequality (x+1)(2x−3)≥ 0.

The solution is x≤−1 or x≥ 32

. In terms of set notation this is denoted by (−∞,−1]∪

[32,∞).

Example 1.1.3 Solve the inequality x(x+2)≥−4.

This is true for any value of x. It is written as R or (−∞,∞) .Something is in the Cartesian product of a set whose elements are sets if it consists

of a single thing taken from each set in the family. Thus (1,2,3) ∈ {1,4, .2}×{1,2,7}×{4,3,7,9} because it consists of exactly one element from each of the sets which are sepa-rated by ×. Also, this is the notation for the Cartesian product of finitely many sets. If Sis a set whose elements are sets, ∏A∈S A signifies the Cartesian product.

The Cartesian product is the set of choice functions, a choice function being a functionwhich selects exactly one element of each set of S . You may think the axiom of choice,stating that the Cartesian product of a nonempty family of nonempty sets is nonempty,is innocuous but there was a time when many mathematicians were ready to throw it outbecause it implies things which are very hard to believe, things which never happen withoutthe axiom of choice.

1.2 The Schroder Bernstein TheoremIt is very important to be able to compare the size of sets in a rational way. The most usefultheorem in this context is the Schroder Bernstein theorem which is the main result to bepresented in this section. The Cartesian product is discussed above. The next definitionreviews this and defines the concept of a function.

Definition 1.2.1 Let X and Y be sets.

X×Y ≡ {(x,y) : x ∈ X and y ∈ Y}

A relation is defined to be a subset of X ×Y . A function f , also called a mapping, is arelation which has the property that if (x,y) and (x,y1) are both elements of the f , theny = y1. The domain of f is defined as

D( f )≡ {x : (x,y) ∈ f} ,