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72 The Nuts and Bolts of Proof, Third Edition
The shaded area in Diagram 3 represents the set A OB:
Diagram 3
EXAMPLE 3. If A, B, and C are any three sets, then:
AC\{B^C)
= {Af^B)\J{Af^ C).
(This is known as the distributive property of the intersection with respect to
the union.)
Proof:
Part 1. An(BUC)^(AnB)U(AnC)
LQtxeAn(BU C). We want to prove that xe(AnB)U(An C).
Because x e An(BUCX then x £ A and x e(BU C).
So, X G ^ and either x e B or x £ C.
Because we know that x e A,WQ can consider two cases: either x e A and
X e B, or X e A and x e C.
Thus, either x e (AnB) or x e (An C).
Therefore, we can conclude that x e(AnB)U(An C).
Part 2. (AnB)U(AnC)£An(BUC)
Let xe(AnB)U(An C). We want to prove that x£An(BUC).
Because xG(AnB)U(An C), then either x e (An B) or x e (An C).
So, either x e A and x e B, or x e A and x e C.
Therefore, in any case x e A and either x e B or x £ C.
Thus, xeA2indxe(BU C).
So, we can conclude that x e An(BUC).
By the conclusions proved in the two preceding parts, we can state that:
An(BuC) = (AnB)u(Anc). •
Let's check the equality An(BU C) = (AnB)U(AnC) using Venn
diagrams.