If U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {x : x is a perfect square less than 10}

B = {x : x is a multiple of 3 less than 10}

Verify (i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′

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Solution

U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {x: x is a perfect square less than 10} = {4, 9}

B = {x: x is a multiple of 3 less than 10} = {3, 6, 9}

(i) A ∪ B = {4, 9} ∪ {3, 6, 9} = {3, 4, 6, 9}

∴ (A ∪ B)′ = U − (A ∪ B) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} − {3, 4, 6, 9} = {0, 1, 2, 5, 7, 8}

A′ = U − A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} − {4, 9} = {0, 1, 2, 3, 5, 6, 7, 8}

B′ = U − B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} − {3, 6, 9} = {0, 1, 2, 4, 5, 7, 8}

∴ A′ ∩ B′ = {0, 1, 2, 3, 5, 6, 7, 8} ∩ {0, 1, 2, 4, 5, 7, 8} = {0, 1, 2, 5, 7, 8}

Hence, (A ∪ B)′ = A′ ∩ B′

(ii) A ∩ B = {4, 9} ∩ {3, 6, 9} = {9}

∴ (A ∩ B)′ = U − (A ∩ B) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} − {9} = {0, 1, 2, 3, 4, 5, 6, 7, 8}

A′ = {0, 1, 2, 3, 5, 6, 7, 8}

B′ = {0, 1, 2, 4, 5, 7, 8}

∴ A′ ∪ B′ = {0, 1, 2, 3, 5, 6, 7, 8} ∪ {0, 1, 2, 4, 5, 7, 8} = {0, 1, 2, 3, 4, 5, 6, 7, 8}

Hence, (A ∩ B)′ = A′ ∪ B′

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