Fundamentals of Probability and Statistics for Engineers

stands for the set of all elements belonging to one or more of the sets Aj,
j 1,2,...,n.Theintersection

is the set of all elements common to all Aj,j 1, 2,...,n. The sets
Aj, j 1, 2,... , n, are disjoint if

U sing Venn diagrams or analytica l procedures, it is easy to verify that union
and intersection operations are associative, commutative, and distributive; that is,

Clearly, we also have

Moreover, the following useful relations hold, all of which can be easily verified
using Venn diagrams:

Basic Probability Concepts 11

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Aj … 2 : 6 †

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…A[B†[CˆA[…B[C†ˆA[B[C;

A[BˆB[A;

…AB†CˆA…BC†ˆABC;

ABˆBA;

A…B[C†ˆ…AB†[…AC†:

… 2 : 8 †

A[AˆAAˆA;

A[;ˆA;

A;ˆ;;

A[SˆS;

ASˆA;

A[AˆS;

AAˆ;:

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A[BˆA[…AB†ˆA‡…AB†;

…A[B†ˆAB;

…AB†ˆA[B;

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Aj

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jˆ 1

Aj;

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