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Question

# (a) If A = {1, 3, 4, 8, 9, 12}, B = {1, 4, 9} and C = {2, 4, 8, 10} Find (i) A ∪ (B ∩ C) (ii) A ∩ (B ∪ C) (iii) (A ∪ B) ∩ (A ∪ C) (iv) (A ∩ B) ∪ (A ∩ C) (b) If A = (2, 4, 6, 8, 10}, B = {1, 2, 3, 4, 5, 6} and C = (1, 3, 5, 7, 9, 11, 13} Verify (i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (iii) (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C) (iv) (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C) (c) If X = {x : x is a prime number less than 12} Y = {x : x is an even number less than 12} Z = {x : x is an odd number less than 12} Show that (i) union of sets of distributive over intersection of sets. (ii) intersection of sets is distributive over union of sets.

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Solution

## (a) A = {1, 3, 4, 8, 9, 12}, B = {1, 4, 9} and C = {2, 4, 8, 10} (i) B ∩ C = {1, 4, 9} ∩ {2, 4, 8, 10} = {4} ∴ A ∪ (B ∩ C) = {1, 3, 4, 8, 9, 12} ∪ {4} = {1, 3, 4, 8, 9, 12} (ii) B ∪ C = {1, 4, 9} ∪ {2, 4, 8, 10} = {1, 2, 4, 8, 9, 10} ∴ A ∩ (B ∪ C) = {1, 3, 4, 8, 9, 12} ∩ {1, 2, 4, 8, 9, 10} = {1, 4, 8, 9} (iii) A ∪ B = {1, 3, 4, 8, 9, 12} ∪ {1, 4, 9} = {1, 3, 4, 8, 9, 12} A ∪ C = {1, 3, 4, 8, 9, 12} ∪ {2, 4, 8, 10} = {1, 2, 3, 4, 8, 9, 10, 12} ∴ (A ∪ B) ∩ (A ∪ C) = {1, 3, 4, 8, 9, 12} ∩ {1, 2, 3, 4, 8, 9, 10, 12} = {1, 3, 4, 8, 9, 12} (iv) A ∩ B = {1, 3, 4, 8, 9, 12} ∩ {1, 4, 9} = {1, 4, 9} A ∩ C = {1, 3, 4, 8, 9, 12} ∩ {2, 4, 8, 10} = {4, 8} ∴ (A ∩ B) ∪ (A ∩ C) = {1, 4, 9} ∪ {4, 8} = {1, 4, 8, 9} (b) A = {2, 4, 6, 8, 10}, B = {1, 2, 3, 4, 5, 6} and C = {1, 3, 5, 7, 9, 11, 13} (i) B ∩ C = {1, 2, 3, 4, 5, 6} ∩ {1, 3, 5, 7, 9, 11, 13} = {1, 3, 5} ∴ A ∪ (B ∩ C) = {2, 4, 6, 8, 10} ∪ {1, 3, 5} = {1, 2, 3, 4, 5, 6, 8, 10} A ∪ B = {2, 4, 6, 8, 10} ∪ {1, 2, 3, 4, 5, 6} = {1, 2, 3, 4, 5, 6, 8, 10} A ∪ C = {2, 4, 6, 8, 10} ∪ {1, 3, 5, 7, 9, 11, 13} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13} ∴ (A ∪ B) ∩ (A ∪ C) = {1, 2, 3, 4, 5, 6, 8, 10} ∩ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13} = {1, 2, 3, 4, 5, 6, 8, 10} Hence, A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (ii) B ∪ C = {1, 2, 3, 4, 5, 6} ∪ {1, 3, 5, 7, 9, 11, 13} = {1, 2, 3, 4, 5, 6, 7, 9, 11, 13} ∴ A ∩ (B ∪ C) = {2, 4, 6, 8, 10} ∩ {1, 2, 3, 4, 5, 6, 7, 9, 11, 13} = {2, 4, 6} A ∩ B = {2, 4, 6, 8, 10} ∩ {1, 2, 3, 4, 5, 6} = {2, 4, 6} A ∩ C = {2, 4, 6, 8, 10} ∩ {1, 3, 5, 7, 9, 11, 13} = ∴ (A ∩ B) ∪ (A ∩ C) = {2, 4, 6} ∪ = {2, 4, 6} Hence, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ B) (iii) (A ∩ B) ∪ C = {2, 4, 6} ∪ {1, 3, 5, 7, 9, 11, 13} = {1, 2, 3, 4, 5, 6, 7, 9, 11, 13} A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13} B ∪ C = {1, 2, 3, 4, 5, 6, 7, 9, 11, 13} ∴ (A ∪ C) ∩ (B ∪ C) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13} ∩ {1, 2, 3, 4, 5, 6, 7, 9, 11, 13} = {1, 2, 3, 4, 5, 6, 7, 9, 11, 13} Hence, (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C) (iv) A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10} ∴ (A ∪ B) ∩ C = {1, 2, 3, 4, 5, 6, 8, 10} ∩ {1, 3, 5, 7, 9, 11, 13} = {1, 3, 5} A ∩ C = B ∩ C = {1, 3, 5} ∴ (A ∩ C) ∪ (B ∩ C) = ∪ {1, 3, 5} = {1, 3, 5} Hence, (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C) (c) X = {x: x is a prime number less than 12} = {2, 3, 5, 7, 11} Y = {x: x is an even number less than 12} = {2, 4, 6, 8, 10} Z = {x: x is an odd number less than 12} = {1, 3, 5, 7, 9, 11} To show: (i) Union of sets is distributive over intersection of sets, i.e., X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z) (ii) Intersection of sets is distributive over union of sets, i.e., X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z) Proof: (i) Y ∩ Z = {2, 4, 6, 8, 10} ∩ {1, 3, 5, 7, 9, 11} = ∴ X ∪ (Y ∩ Z) = {2, 3, 5, 7, 11} ∪ = {2, 3, 5, 7, 11} X ∪ Y = {2, 3, 5, 7, 11} ∪ {2, 4, 6, 8, 10} = {2, 3, 4, 5, 6, 7, 8, 10, 11} X ∪ Z = {2, 3, 5, 7, 11} ∪ {1, 3, 5, 7, 9, 11} = {1, 2, 3, 5, 7, 9, 11} ∴(X ∪ Y) ∩ (X ∪ Z) = {2, 3, 4, 5, 6, 7, 8, 10, 11} ∩ {1, 2, 3, 5, 7, 9, 11} = {2, 3, 5, 7, 11} Hence, X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z) (ii) Y ∪ Z = {2, 4, 6, 8, 10} ∪ {1, 3, 5, 7, 9, 11} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} ∴ X ∩ (Y ∪ Z) = {2, 3, 5, 7, 11} ∩ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}∴ = {2, 3, 5, 7, 11} X ∩ Y = {2, 3, 5, 7, 11} ∩ {2, 4, 6, 8, 10} = {2} X ∩ Z = {2, 3, 5, 7, 11} ∩ {1, 3, 5, 7, 9, 11} = {3, 5, 7, 11} ∴ (X ∩ Y) ∪ (X ∩ Z) = {2} ∪ {3, 5, 7, 11} = {2, 3, 5, 7, 11} Hence, X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z)

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