Page 28 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 11

stands for the set of all elements belonging to one or more of the sets A j ,
ˆ
j 1, 2,..., n. The intersection
n
\
A 1 A 2 ... A n ˆ A j 2:6†
jˆ1
is the set of all elements common to all A j , j ˆ 1, 2, ..., n. The sets
A j , j ˆ 1, 2, . . . , n, are disjoint if

A i A j ˆ ;; for every i; j i 6ˆ j†: 2:7†
Using Venn diagrams or analytical procedures, it is easy to verify that union
and intersection operations are associative, commutative, and distributive; that is,
9
A [ B†[ C ˆ A [ B [ C†ˆ A [ B [ C; >
>
A [ B ˆ B [ A; >
>
>
=
AB†C ˆ A BC†ˆ ABC; 2:8†
AB ˆ BA; >
>
>
>
>
A B [ C†ˆ AB†[ AC†: ;
Clearly, we also have
A [ A ˆ AA ˆ A; 9
>
A [; ˆ A; >
>
>
>
>
A;ˆ;; >
>
>
=
A [ S ˆ S; 2:9†
AS ˆ A; >
>
>
>
>
A [ A ˆ S; >
>
>
>
;
AA ˆ;:
Moreover, the following useful relations hold, all of which can be easily verified
using Venn diagrams:
9
A [ BC†ˆ A [ B† A [ C†;
>
>
>
>
>
A [ B ˆ A [ AB†ˆ A ‡ AB†; >
>
>
>
A [ B†ˆ A B; >
>
>
>
>
>
AB†ˆ A [ B; >
>
=
2:10†
!
n n
[ \ >
ˆ A j ; >
>
A j >
>
jˆ1 jˆ1 >
>
>
>
>
n ! n >
>
>
\ [ >
A j ˆ A j : >
>
>
;
jˆ1 jˆ1
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