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• The Duality principle states that: If in an identity, we replace unions
by intersections, intersections by unions, S by ∅, and ∅ by S, then
the identity is preserved.

THEOREM 1
If we define the difference of two events A – A to mean the events in which
2
1
A occurs but not A , the following equalities are valid:
2
1
PA( − A ) = P A( ) − P A( ∩ A ) (10.24)
1 2 1 1 2
PA( − A ) = P A( ) − P A( ∩ A ) (10.25)
2 1 2 1 2

PA( ∪ A ) = P A( ) + P A( ) − P A( ∩ A ) (10.26)
1 2 1 2 1 2
PROOF From the basic set theory algebra results, we can deduce the follow-
ing equalities:

A = ( A − A ∪) ( A ∩ A ) (10.27)
1 1 2 1 2
A = ( A − A ∪) ( A ∩ A ) (10.28)
2 2 1 1 2
A ∪ A = ( A − A ∪) ( A − A ∪) ( A ∩ A ) (10.29)
1 2 1 2 2 1 1 2
Further note that the events (A – A ), (A – A ), and (A ∩ A ) are mutually
1
2
1
1
2
2
exclusive. Using the results from Pb. 10.7, Eqs. (10.27) and (10.28), and the
preceding comment, we can write:
PA( ) = PA( − A ) + P A( ∩ A ) (10.30)
1 1 2 1 2
PA( ) = PA( − A ) + P A( ∩ A ) (10.31)
2 2 1 1 2
which establish Eqs. (10.24) and (10.25). Next, consider Eq. (10.29); because of
the mutual exclusivity of each event represented by each of the parenthesis
on its LHS, we can use the results of Pb. 10.7, to write:

PA( ∪ A ) = P A( − A ) + PA( − A ) + P A( ∩ A ) (10.32)
1 2 1 2 2 1 1 2
using Eqs. (10.30) and (10.31), this can be reduced to Eq. (10.26).

THEOREM 2
Given any n events A , A , …, A and defining P , P , P , …, P to mean:
2
3
n
n
2
1
1
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