Page 10 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 1) PROBABILITY

The set containing no element is called the null set, denoted 0. Note that

6. Disjoint Sets:
Two sets A and B are called disjoint or mutually exclusive if they contain no common element,
that is, if A n B = 0.
The definitions of the union and intersection of two sets can be extended to any finite number of
sets as follows:
n
UA~=A, u A,U..-U A,
i= 1
= ([: [E Al or [E AZ or .-- E A,)

= (5: 5 E Al and 5 E A, and 5 E A,)
Note that these definitions can be extended to an infinite number of sets:

In our definition of event, we state that every subset of S is an event, including S and the null set
0. Then
S = the certain event
@ = the impossible event

If A and B are events in S, then
2 = the event that A did not occur
A u B = the event that either A or B or both occurred
A n B = the event that both A and B occurred
Similarly, if A,, A,, . . . , A, are a sequence of events in S, then
n
U A, = the event that at least one of the A, occurred;
i= 1
n
n Ai = the event that all of the A, occurred.
i= 1

B. Venn Diagram:

A graphical representation that is very useful for illustrating set operation is the Venn diagram.
For instance, in the three Venn diagrams shown in Fig. 1-1, the shaded areas represent, respectively,
the events A u B, A n B, and A. The Venn diagram in Fig.. 1-2 indicates that B c A and the event
A n B is shown as the shaded area.