Q. 5

# 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’)

Answer :

(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]

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