- Review construction of a three set Venn diagram
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U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
A = {1,4,8,9,10,12}
B = {2,4,5,9,10,13}
C = {1,3,4,8,9,11}
- Is A' ∪ B = A' ∩ B
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Find the regions for A' ∪ B
A' is composed of regions III, VI, VII and VIII
B is composed of regions II, III, V, VI
A' ∪ B is composed of regions II, III, V, VI, VII, VIII
Find the regions for A' ∩ B
A' is composed of regions III, VI, VII and VIII
B is composed of regions II, III, V, VI
A' ∩ B is composed of regions III, VI
Since these are not the same regions, the sets are not equal in general.
When would they be equal?
- Is (A ∪ B)' = A' ∩ B'
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(A ∪ B)'
A is composed of regions I, II, IV, and V
B is composed of regions II, III, V, VI
A ∪ B is composed of regions I, II, III, IV, V, VI
(A ∪ B)' is composed of regions VII and VIII
A' ∩ B'
A' is composed of regions III, VI, VII, VIII
B' is composed of regions I, IV, VII and VIII
A' ∩ B' is composed of regions VII and VIII
Since these are equal, the sets are equal in general.
- This is a special case, one of two which make up De Morgan's Laws
- A' ∪ B' = (A ∩ B)'
- A' ∩ B' = (A ∪ B)'
- We will see De Morgan again in logic