Let A= The set of all natural numbers less than 8, B=The set of all prime numbers less than 8, C=The set of even prime number. Verify that (A∩B) C= (AC)∩(BC).
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Answer:
A = {1, 2, 3, 4, 5, 6, 7} B = {2, 3, 5, 7} C = {2} (i) (A ∩ B) x C = (A x C) ∩ (B x C) A ∩ B = {1, 2, 3, 4, 5, 6, 7} ∩ {2, 3, 5, 7} = {2, 3, 5, 7} (A ∩ B) x C = {2, 3, 5, 7} x {2} = {(2, 2) (3, 2) (5, 2) (7, 2)} … (1) A x C = {1,2, 3, 4, 5, 6, 7} x {2} = {(1,2) (2, 2) (3, 2) (4, 2) (5.2) (6, 2) (7, 2)} B x C = {2, 3, 5, 7} x {2} = {(2, 2) (3, 2) (5, 2) (7, 2)} (A x C) ∩ (B x C) = {(2, 2) (3, 2) (5, 2) (7, 2)} … (2) From (1) and (2) we get (A ∩ B) x C = (A x C) ∩ (B x C) (ii) A x (B – C) = (A x B) – (A x C) B – C = {2, 3, 5, 7} – {2} = {3,5,7} A x (B – C) = {1,2, 3, 4, 5, 6,7} x {3,5,7} = {(1, 3) (1, 5) (1, 7) (2, 3) (2, 5) (2, 7) (3, 3) (3, 5) (3, 7) (4, 3) (4, 5) (4, 7) (5, 3) (5, 5) (5, 7) (6, 3) (6, 5) (6, 7) (7, 3) (7, 5) (7, 7)} … (1) A x B = {1,2, 3, 4, 5, 6, 7} × {2, 3, 5,7} = {(1, 2) (1, 3) (1, 5) (1, 7) (2, 2) (2, 3) (2, 5) (2, 7) (3, 2) (3, 3) (3, 5) (3, 7) (4, 2) (4, 3) (4, 5) (4, 7) (5, 2) (5, 3) (5, 5) (5, 7) (6, 2) (6, 3) (6, 5) (6, 7) (7, 2) (7, 3) (7, 5) (7, 7)} A x C = {1,2, 3,4, 5, 6, 7} x {2} = {(1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6.2) (7,2)} (A x B) – (A x C) = {(1, 3) (1, 5) (1, 7) (2, 3) (2, 5) (2, 7) (3, 3) (3, 5) (3, 7) (4, 3) (4, 5) (4, 7) (5, 3) (5, 5) (5, 7) (6, 3) (6, 5) (6, 7) (7, 3) (7, 5) (7, 7)} … (2) From (1) and (2) we get A x (B – C) = (A x B) – (A x C)
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