Page 137 - Discrete Mathematics and Its Applications

116 2 / Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

collections in an organized fashion. We now provide a definition of a set. This definition is an
intuitive definition, which is not part of a formal theory of sets.

DEFINITION 1 A set is an unordered collection of objects, called elements or members of the set. A set is
said to contain its elements. We write a ∈ A to denote that a is an element of the set A. The
notation a ∈ A denotes that a is not an element of the set A.

It is common for sets to be denoted using uppercase letters. Lowercase letters are usually
used to denote elements of sets.
There are several ways to describe a set. One way is to list all the members of a set, when
this is possible. We use a notation where all members of the set are listed between braces. For
example, the notation {a, b, c, d} represents the set with the four elements a, b, c, and d. This
way of describing a set is known as the roster method.

EXAMPLE 1 The set V of all vowels in the English alphabet can be written as V ={a, e, i, o, u}. ▲

EXAMPLE 2 The set O of odd positive integers less than 10 can be expressed by O ={1, 3, 5, 7, 9}. ▲

EXAMPLE 3 Although sets are usually used to group together elements with common properties, there is
nothing that prevents a set from having seemingly unrelated elements. For instance, {a, 2, Fred,
New Jersey} is the set containing the four elements a, 2, Fred, and New Jersey. ▲

Sometimes the roster method is used to describe a set without listing all its members. Some
members of the set are listed, and then ellipses (...) are used when the general pattern of the
elements is obvious.

EXAMPLE 4 The set of positive integers less than 100 can be denoted by {1, 2, 3,..., 99}. ▲
Another way to describe a set is to use set builder notation. We characterize all those
elements in the set by stating the property or properties they must have to be members. For
instance, the set O of all odd positive integers less than 10 can be written as

O ={x | x is an odd positive integer less than 10},

or, specifying the universe as the set of positive integers, as

+
O ={x ∈ Z | x is odd and x< 10}.
We often use this type of notation to describe sets when it is impossible to list all the elements
+
of the set. For instance, the set Q of all positive rational numbers can be written as
p
+
Q ={x ∈ R | x = , for some positive integers p and q}.
q
Beware that mathe- These sets, each denoted using a boldface letter, play an important role in discrete mathe-
maticians disagree
matics:
whether 0 is a natural
number. We consider it N ={0, 1, 2, 3,...}, the set of natural numbers
quite natural.
Z ={..., −2, −1, 0, 1, 2,...}, the set of integers
+
Z ={1, 2, 3,...}, the set of positive integers
Q ={p/q | p ∈ Z,q ∈ Z, and q = 0}, the set of rational numbers
R, the set of real numbers
R , the set of positive real numbers
+
C, the set of complex numbers.