Let A = {x ∈ W | x < 2}, B = {x ∈ N | 1 < x < 4} and C = {3, 5}. Verify that (i) A x (B ∪ C) = (A x B) ∪ (A x C)

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answered Oct 9, 2020 by Aanchi (47.6k points)
selected Oct 9, 2020 by Darshee

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A = {x ∈ W | x < 2} = {0,1}

B = {x ∈ N | 1 < x < 4} = {2,3,4}

C = {3,5}

(i) LHS = A x (B ∪ C)

B ∪ C = {2, 3, 4} ∪ {3, 5}

= {2, 3, 4, 5}

A x (B ∪ C) = {(0, 2), (0, 3), (0,4), (0, 5), (1, 2) , (1, 3), (1, 4),(1, 5)} … (1)

RHS = (A x B) ∪ (A x C)

(A x B) = {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)}

(A x C) = {(0, 3), (0, 5), (1, 3), (1, 5)}

(A x B) ∪ (A x C) = {(0, 2), (0, 3), (0,4), (1, 2), (1, 3), (1, 4), (0, 5), (1, 5)} … (2)

(1) = (2),

LHS = RHS

Hence it is proved.

(ii) A x (B ∩ C) = (A x B) ∩ (A x C)

LHS = A x (B ∩ C)

(B ∩ C) = {3}

A x (B ∩ C) = {(0, 3), (1, 3)} … (1)

RHS = (A x B) ∩ (A x C)

(A x B) = {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)}

(A x C) = {(0, 3), (0, 5), (1, 3), (1, 5)}

(A x B) ∩ (A x C) = {(0, 3), (1, 3)} ... (2)

(1) = (2)

⇒ LHS = RHS.

Hence it is verified.

(iii) (A ∪ B) x C = (A x C) ∪ (B x C)

LHS = (A ∪ B) x C

A ∪ B = {0, 1, 2, 3, 4}

(A ∪ B) x C = {(0, 3), (0, 5), (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} … (1)

RHS = (A x C) ∪ (B x C) (A x C) = {(0, 3), (0, 5), (1, 3), (1, 5)}

(B x C) = {(2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)}

(A x C) ∪ (B x C) = {(0, 3), (0, 5), (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} … (2)

(1) = (2)

∴ LHS = RHS. Hence it is verified.