If A = {x : x ϵ N, x ≤ 7}, B = {x : x is prime, x < 8} and C = {x : x ϵ N, x is odd and x < 10}, verify that:

(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

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: Natural numbers start from 1

A = {1, 2, 3, 4, 5, 6, 7}

B = {2, 3, 5, 7}

C = {1, 3, 5, 7, 9}

(i) B∩C = {3, 5, 7}

AU(B∩C) = {1, 2, 3, 4, 5, 6, 7}

A∪B = {1, 2, 3, 4, 5, 6, 7}

A∪C = {1, 2, 3, 4, 5, 6, 7, 9}

(A∪B)∩(A∪C) = {1, 23, 4, 5, 6, 7}

A∪(B∩C) = (A∪B)∩(A∪C)

Hence proved

(ii) B∪C = {1, 2, 3, 5, 7, 9}

A∩(B∪C) = {1, 2, 3, 5, 7}

A∩B = {2, 3, 5, 7}

A∩C = {1, 3, 5, 7}

(A∩B)∪(A∩C) = {1, 2, 3, 5, 7}

A∩(B∪C) = (A∩B)∪(A∩C)