Page 26 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 9

B
A

Figure 2.1 Venn diagram for A B

Example 2.1. Let A ˆf2, 4g and B ˆf1, 2, 3, 4g Then A B, since every
element of A is also an element of B . This relationship can also be presented
graphically by using a Venn diagram, as shown in Figure 2.1. The set B
occupies the interior of the larger circle and A the shaded area in the figure.

It is clear that an empty set is a subset of any set. When both A B and
B A , set A is then equal to B , and we write

A ˆ B: 2:3†

We now give meaning to a particular set we shall call space. In our develop-
ment, we consider only sets that are subsets of a fixed (nonempty) set. This
‘largest’ set containing all elements of all the sets under consideration is called
space and is denoted by the symbol S.
Consider a subset A in S. The set of all elements in S that are not elements of
A is called the complement of A, and we denote it by A. A Venn diagram
showing A and A is given in Figure 2.2 in which space S is shown as a rectangle
and A is the shaded area. We note here that the following relations clearly hold:

S ˆ;; ;ˆ S; A ˆ A: 2:4†

2.1.1 SET OPERATIONS

Let us now consider some algebraic operations of sets A, B, C ,. . . that are
subsets of space S.
The union or sum of A and B, denoted by A [ B , is the set of all elements
belonging to A or B or both.

S A A

Figure 2.2 A and A

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