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The following set properties are
given here in preparation for the properties for addition and
multiplication in arithmetic. Note the close similarity between
these properties and their corresponding properties for addition
and multiplication.
Distributive
Properties:
The
Distributive Property of
Union over Intersection and the
Distributive Property of
Intersection over Union show two ways of finding results for
certain problems mixing the set operations of union and
intersection.
General Property:
A
∪
(B
∩ C) = (A
∪
B)
∩ (A
∪
C) and
A
∩ (B
∪
C) = (A
∩ B)
∪
(A
∩ C)
Example: Let A
= {a, n, t}, B = {t,
a, p}, and C = {s, a,
p}. Then
A
∪
(B
∩ C) = {a, n, t}
∪
{a, p}
= {p, a, n, t} = {p, a, n, t}
∩ {p, a,
n, t, s} = (A
∪
B)
∩ (A
∪
C)
A
∩ (B
∪
C)
= {a, n, t}
∩ {t, a, p, s} = {a, t} = {a, t}
∪
{a} = (A
∩ B)
∪
(A
∩ C)