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Mathematics

If A = {x ∈ W : 5 < x < 10}, B = {3, 4, 5, 6, 7} and C = {x = 2n ; n ∈ N and n ≤ 4}. Find :

(i) A ∩ (B ∪ C)

(ii) (B ∪ A) ∩ (B ∪ C)

(iii) B ∪ (A ∩ C)

(iv) (A ∩ B) ∪ (A ∩ C)

Name the sets which are equal.

Sets

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Answer

A = {x ∈ W : 5 < x < 10}

A = {6, 7, 8, 9}

B = {3, 4, 5, 6, 7}

C = {x = 2n ; n ∈ N and n ≤ 4}

C = {2, 4, 6, 8}

(i) B ∪ C - contains all the elements in set B and C.

B ∪ C = {3, 4, 5, 6, 7} ∪ {2, 4, 6, 8}

B ∪ C = {2, 3, 4, 5, 6, 7, 8}

A ∩ (B ∪ C) - contains all the common elements in set A and (B ∪ C).

A ∩ (B ∪ C) = {6, 7, 8, 9} ∩ {2, 3, 4, 5, 6, 7, 8}

A ∩ (B ∪ C) = {6, 7, 8}

(ii) B ∪ A - contains all the elements in set B and A.

B ∪ A = {3, 4, 5, 6, 7} ∪ {6, 7, 8, 9}

B ∪ A = {3, 4, 5, 6, 7, 8, 9}

B ∪ C - contains all the elements in set B and C.

B ∪ C = {3, 4, 5, 6, 7} ∪ {2, 4, 6, 8}

B ∪ C = {2, 3, 4, 5, 6, 7, 8}

(B ∪ A) ∩ (B ∪ C) - contains all common elements in set (B ∪ A) and (B ∪ C).

(B ∪ A) ∩ (B ∪ C) = {3, 4, 5, 6, 7, 8, 9} ∩ {2, 3, 4, 5, 6, 7, 8}

(B ∪ A) ∩ (B ∪ C) = {3, 4, 5, 6, 7, 8}

(iii) A ∩ C - contains all the common elements in set A and C.

A ∩ C = {6, 7, 8, 9} ∩ {2, 4, 6, 8}

A ∩ C = {6, 8}

B ∪ (A ∩ C) - contains all the elements of B and (A ∩ C).

B ∪ (A ∩ C) = {3, 4, 5, 6, 7} ∪ {6, 8}

B ∪ (A ∩ C) = {3, 4, 5, 6, 7, 8}

(iv) A ∩ B - contains all the common elements in set A and B.

A ∩ B = {6, 7, 8, 9} ∩ {3, 4, 5, 6, 7}

A ∩ B = {6, 7}

A ∩ C - contains all the common elements in set A and C.

A ∩ C = {6, 7, 8, 9} ∩ {2, 4, 6, 8}

A ∩ C = {6, 8}

(A ∩ B) ∪ (A ∩ C) - contains all the elements of (A ∩ B) and (A ∩ C).

(A ∩ B) ∪ (A ∩ C) = {6, 7} ∪ {6, 8}

(A ∩ B) ∪ (A ∩ C) = {6, 7, 8}

As we can see that some sets are having same elements i.e, they are equal.

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and B ∪ (A ∩ C) = (B ∪ A) ∩ (B ∪ C)

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