Basic Definitions 5Definition 1. Let A and B be sets. Then, A = B or A is equal to B if both A and B have
the same elements.The word if has a special meaning when used in definitions. Definition 1 states that
A = B if A and B have the same elements. Since the word if is inside a definition, it is
implied that A A B whenever the condition is not satisfied. Thus, "A = B" is just a short
way to say "A and B have the same elements."Example 2.(a) {nEZ:n^2 - n-2=01={nEZ:(n-2)(n+1)=0}={2,--1}.
(b) {n ENn: 2 n-2=0}={2}because-1 0lN.
(c) {x N: (x + 1)2 - (x - 1)2 -4x =0) =Nbecause
(x+1)2 -(x-1)2 -4x=x^2 +2x+l-x^2 +2x-l-4x
=0is an algebraic identity (true for all x).The empty set 0 can be described in several ways; for example,{x E N : x <x).
The set of continents south of Antarctica.
The set of round squares.Why is {x e N : x < xJ equal to the set of round squares? We know that if two sets
are equal, then they must have the same elements. So, if{x E N : x < x} A the set of round squaresthen there is an element in one of these sets that is not in the other set. This cannot be true,
however, since neither set has any elements at all.1.1.4 Finite and Infinite Sets
Some sets, like {0, 1, 2, 3}, have the property that a person could list their elements
and finish listing them. We can describe this condition a little more formally: Ei-
ther the set has no elements, or its elements can be matched with the elements of
some subset {1, 2 .... n} of the natural numbers. Such sets are called a finite set. So,{Liverpool, London, Los Angeles) is a finite set-that is, elements could be matched as
1 (Liverpool), 2 (London), and 3 (Los Angeles)The empty set 0 has zero elements, so it is also finite.
Some sets are infinite sets, or not finite sets, like Z, R•, and N'. There is no way to
match all the elements of Z with a set {1, 2, ..... n for any fixed n.
1.1.5 Relations Between Sets
Besides equality, another important relation between sets occurs when all the elements of
one set are also elements of a second set.