Fundamentals of Probability and Statistics for Engineers

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

It is clear that an empty set is a subset of any set. When both and

, set is then equal to , and we write

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 byA. A Venn diagram
showing A andA is given in Figure 2.2 in which space S is shown as a rectangle
andA is the shaded area. We note here that the following relations clearly hold:

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 , is the set of all elements
belonging to A or B or both.

B

A

Figure 2.1 Venn diagram for

S A A

Figure 2. 2 AandA

Basic Probability Concepts 9

Aˆf2, 4g Bˆf1, 2, 3, 4g AB,
A B
B
A
AB
BA A B

AˆB: … 2 : 3 †

AB

Sˆ;; ;ˆS; AˆA:… 2 : 4 †

A[B