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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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