Let A = {2, 3, 5, 7, 11, 13}, B = {5, 7, 9, 11, 15} be subsets of U = {2, 3, 5, 7, 9, 11, 13, 15}.
Using Venn diagrams, verify that:
(i) (A ∪ B’) = (A’ ∩ B’)
(ii) (A ∩ B)’ = (A’ ∪ B’)
(i) Here blue region denotes set A - B
The green region denotes set B - A
The overlapping region denotes AB, and the orange region denotes the universal set U.
From the Venn diagram we get (AB’) = {2, 3, 5, 7, 11, 13} (B’ is the set excluding those elements present in set B i.e. A - B region)
A’ = {9, 15} and B’ = {2, 3, 13}
Therefore A’B’ = {}
Therefore (A ∪ B’) (A’ ∩ B’) [Verified]
(ii) From the Venn diagram we get (A ∩ B)’ = {2, 3, 9, 13, 15} (elements except those present in A ∩ B)
(A’ ∪ B’) = {2, 3, 9, 13, 15}
Therefore, (A ∩ B)’ = (A’ ∪ B’) [Verified]