Question 35 of 47intermediate🔧 ApplyShort Answer2 marks

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that

(i)

(A ∪ B)' = A' ∩ B'

Answer

Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}.

Finding A' and B': A' = U - A = {1, 3, 5, 7, 9} B' = U - B = {1, 4, 6, 8, 9}

Finding LHS: A ∪ B = {2, 3, 4, 5, 6, 7, 8} (A ∪ B)' = {1, 9}

Finding RHS: A' ∩ B' = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} = {1, 9}

Since LHS = RHS = {1, 9}, we have verified that (A ∪ B)' = A' ∩ B'.

(ii)

(A ∩ B)' = A' ∪ B'

Answer

Finding LHS: A ∩ B = {2} (A ∩ B)' = {1, 3, 4, 5, 6, 7, 8, 9}

Finding RHS: A' ∪ B' = {1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}

Since LHS = RHS = {1, 3, 4, 5, 6, 7, 8, 9}, we have verified that (A ∩ B)' = A' ∪ B'.

Explanation

This question verifies De Morgan's Laws. For part (i), we find the complement of the union equals the intersection of complements. For part (ii), we find the complement of the intersection equals the union of complements. Both are verified by computing each side separately and showing equality.

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