Sixteen things you can say about A and B
Each of the 16 sets below is indicated by a shaded region.
A
B
A
A
A
A
B
B
if A then B
Ac ∪ B
A
A
B
not A
Ac
A or B
A∪B
A
B
B
A
B
B
B
neither A nor B
(A ∪ B)c
A
B
A and not B
A ∩ Bc
A
A
B
A
not B
Bc
B
A and B
A∩B
A
if B then A
A ∪ Bc
A if and only if B
(A ∩ B) ∪ (Ac ∩ B c )
A or B but not both
(A ∩ B c ) ∪ (Ac ∩ B)
Complementary Laws:
Associative Laws:
Commutative Laws:
Distributive Laws:
De Morgan’s Laws:
A ∪ Ac = ∅c
A ∪ (B ∪ C) = (A ∪ B) ∪ C
A∪B =B∪A
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(A ∪ B)c = Ac ∩ B c
A
B
not both A and B
(A ∩ B)c
B
A
B
B
true
A ∪ Ac
A
B
B and not A
Ac ∩ B
A
B
false
A ∩ Ac
A ∩ Ac = ∅
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A∩B =B∩A
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
(A ∩ B)c = Ac ∪ B c
MIT OpenCourseWare
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Combinatorics: The Fine Art of Counting
Summer 2007
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