2 CHAPTER 1 Sets, Proof Templates, and Induction
is usually called a stamp collection. The set of past presidents of the United States consists
of
fGeorge Washington, John Adams, Thomas Jefferson, ...
The "..." is called an ellipsis and indicates that the list contains other elements.
What is the basic characteristic of a set? For any set A and any element b, either b is
in A or b is not in A. If you ask whether an element is in a set, the answer is either yes or
no.
Is 0 in {1, 21? No.
Is 0 in {0, 1, 2, 3, 4, 5, 1003456792311? Yes.
Is New York in {Liverpool, London, Los Angeles I? No.
Is green in {red, yellow, blue)? No.
Is New York in {England, France, United States)? No.
In mathematical terminology, 0 is an element of {0, 1, 2, 3, 4, 5, 1003456792311, and
green is not an element of {red, yellow, blue).
The expression "is an element of" is denoted by the symbol E, a form of the Greek
letter epsilon. For example, we write0 E {0, 1,2,3,4,5, 1003456792311
and
green 0 {red, blue, yellow)
The slash through the E symbol means not, just as it does in A. Is a member of, is con-
tained in, or simply is in means the same as "is an element of." Mathematics, like ordinary
language, is full of synonyms.
Despite its frequent use, the term set is not defined in terms of other concepts. Like
the terms point and line in plane geometry, set is a primitive concept. Just assume there are
elements, there are sets, and that for a set A and an element b, the assertion b E A is either
true or false.
An important distinction needs to be made between I and I 1}. They are not the same.
By itself, I is a number but not a set, and { 11 is the set containing the element 1. Similarly,
{1} and {{1}} are not the same thing: 1 E {1}, but {1) 0 {1}. So, also, {1} E {{1)), but 1 V
I{l1}. Similarly, the set {1, 2) has two elements, 1 and 2, but {{1, 2)} has only one element,
{1, 2).
The number of times an element is listed and the order in which elements are listed
are both unimportant. For example, the elements of {2, 3} are 2 and 3. The elements of
{3, 2, 21 are also 2 and 3. Consequently, these two sets contain the same elements. That is,
these two sets are equal, as we shall see later.1.1.1 Describing Sets Mathematically
We present three different methods to describe a set. The first is by a list of all the elements.
The second is by a description of some property the elements have. The third is by a
description based on some other sets. In all these methods, we use the symbols { and I
to indicate that a set is being defined. The "language" used to describe sets is called set-
theoretic notation.