Given A = {0, 1, 2, 4, 5}, B = {0, 2, 4, 6, 8} and C = {0, 3, 6, 9}. Show that :

(i) A ∪ (B ∪ C) = (A ∪ B) ∪ C, i.e., the union of sets is associative.

(ii) A ∩ (B ∩ C) = (A ∩ B) ∩ C, i.e., the intersection of sets is associative.

A = {0, 1, 2, 4, 5}

B = {0, 2, 4, 6, 8}

C = {0, 3, 6, 9}

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

B ∪ C = {0, 2, 4, 6, 8} ∪ {0, 3, 6, 9}

B ∪ C = {0, 2, 3, 4, 6, 8, 9}

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

A ∪ (B ∪ C) = {0, 1, 2, 4, 5} ∪ {0, 2, 3, 4, 6, 8, 9}

A ∪ (B ∪ C) = {0, 1, 2, 3, 4, 5, 6, 8, 9} ……………(1)

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

A ∪ B = {0, 1, 2, 4, 5} ∪ {0, 2, 4, 6, 8}

A ∪ B = {0, 1, 2, 4, 5, 6, 8}

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

(A ∪ B) ∪ C = {0, 1, 2, 4, 5, 6, 8} ∪ {0, 3, 6, 9}

(A ∪ B) ∪ C = {0, 1, 2, 3, 4, 5, 6, 8, 9} ……………(2)

Hence from (1) and (2), all the elements of A ∪ (B ∪ C) and (A ∪ B) ∪ C are same.

A ∪ (B ∪ C) = (A ∪ B) ∪ C, i.e. the union of sets is associative.

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

B ∩ C = {0, 2, 4, 6, 8} ∩ {0, 3, 6, 9}

B ∩ C = {0,6}

A ∩ (B ∩ C) = {0, 1, 2, 4, 5} ∩ {0,6}

A ∩ (B ∩ C) = {0} ……………(3)

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

A ∩ B = {0, 1, 2, 4, 5} ∩ {0, 2, 4, 6, 8}

A ∩ B = {0, 2, 4}

(A ∩ B) ∩ C = {0, 2, 4} ∩ {0, 3, 6, 9}

(A ∩ B) ∩ C = {0} ……………(4)

Hence from (3) and (4), all the elements of A ∩ (B ∩ C) and (A ∩ B) ∩ C are same.

A ∩ (B ∩ C) = (A ∩ B) ∩ C, i.e., the intersection of sets is associative.