Verify the associative property of intersection of sets for A = { - 11 , 2 , 5 , 7 } B = { 3 , 5 , 6 , 13 } and C = { 2 , 3 , 5 , 9 } .

We need to verify, A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C So, B ∩ C = { 3 , 5 , 6 , 13 } ∩ { 2 , 3 , 5 , 9 } = { 3 , 5 } A ∩ ( B ∩ C ) = { - 11 , 2 , 5 , 7 } ∩ { 3 , 5 } = { 5 } . . . . ( i ) Similarly, A ∩ B = { - 11 , 2 , 5 , 7 } ∩ { 3 , 5 , 6 , 13 } = { 5 } ( A ∩ B ) ∩ C = { 5 } ∩ { 2 , 3 , 5 , 9 } = { 5 } . . . . ( i i ) From ( i ) and ( i i ) , we get A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C Hence, intersection of given sets is associative.

Verify the associative property of intersection of sets for A = { - 11 , 2 , 5 , 7 } B = { 3 , 5 , 6 , 13 } and C = { 2 , 3 , 5 , 9 } .

We need to verify, A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C So, B ∩ C = { 3 , 5 , 6 , 13 } ∩ { 2 , 3 , 5 , 9 } = { 3 , 5 } A ∩ ( B ∩ C ) = { - 11 , 2 , 5 , 7 } ∩ { 3 , 5 } = { 5 } . . . . ( i ) Similarly, A ∩ B = { - 11 , 2 , 5 , 7 } ∩ { 3 , 5 , 6 , 13 } = { 5 } ( A ∩ B ) ∩ C = { 5 } ∩ { 2 , 3 , 5 , 9 } = { 5 } . . . . ( i i ) From ( i ) and ( i i ) , we get A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C Hence, intersection of given sets is associative.

If A = x : x = 2 n , n ∈ W and n < 4 } , B = { x : x = 2 n , n ∈ ℕ and n ≤ 4 } and C = { 0 , 1 , 2 , 5 , 6 } , then verify the associative property of intersection of sets.

From the given conditions, A = x : x = 2 n , n ∈ W , n < 4 ⇒ x = 2 ° = 1 x = 2 1 = 2 x = 2 2 = 4 x = 2 3 = 8 ∴ A = { 1 , 2 , 4 , 8 } Similarly, B = { x : x = 2 n , n ∈ N and n ≤ 4 } ⇒ x = 2 × 1 = 2 x = 2 × 2 = 4 x = 2 × 3 = 6 x = 2 × 4 = 8 ∴ B = { 2 , 4 , 6 , 8 } Given, C = { 0 , 1 , 2 , 5 , 6 } According to the associative property of intersection of sets, we can write A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C Then, B ∩ C = { 2 , 6 } A ∩ ( B ∩ C ) = { 1 , 2 , 4 , 8 } ∩ { 2 , 6 } = { 2 } . . . . . . . . . . . ( i ) A ∩ B = { 1 , 2 , 4 , 8 } ∩ { 2 , 4 , 6 , 8 } = { 2 , 4 , 8 } ( A ∩ B ) ∩ C = { 2 , 4 , 8 } ∩ { 0 , 1 , 2 , 5 , 6 } = { 2 } . . . . . . . ( i i ) From ( i ) and ( i i ) , we conclude, A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C Hence, intersection of given sets is associative.

Verify, ( A - B ) ∪ ( A - C ) ∪ ( A ∩ B ) = A .

Let A = x , y , B = y , z and C = z , x Then, A - B = x and A - C = y Similarly, A ∩ B = y Therefore, ( A - B ) ∪ ( A - C ) ∪ ( A ∩ B ) = x ∪ y ∪ y ⇒ ( A - B ) ∪ ( A - C ) ∪ ( A ∩ B ) = x , y Hence, ( A - B ) ∪ ( A - C ) ∪ ( A ∩ B ) = A .

Verify, A - B = A ∩ B ' .

Let U = 1 , 2 , 3 , 4 , 5 , 6 , 7 , A = 1 , 4 and B = 4 , 5 , 7 Then, A - B = 1 and B ' = U - B = 1 , 2 , 3 , 6 So, A ∩ B ' = 1 , 4 ∩ 1 , 2 , 3 , 6 = 1 Hence, A - B = A ∩ B ' .

Verify, ( A - B ) ∪ B - A ' = A

Let U = x , y , z , p , A = x , y and B = y , z Then, A - B = x and A ' = U - A = x , y , z , p - x , y = z , p So, B - A ' = y Thus, A - B ∪ B - A ' = x ∪ y = x , y Hence, ( A - B ) ∪ B - A ' = A

For any two sets A and B , verify A ∩ ( A ∪ B ) ' = .

Let U = x , y , z , p , A = x , y and B = y , z Now, A ∪ B = { x , y , z } Then, A U B ' = U - A U B = x , y , z , p - x , y , z = p Hence, A ∩ A ∪ B ' = x , y ∩ p = .

For any two sets A and B , verify ( A ∪ B ) ' ∪ A ' ∩ B = A ' .

Let U = 1 , 2 , 3 , 4 , 5 , 6 , A = 1 , 2 and B = 4 , 5 Then, A ∪ B = 1 , 2 , 4 , 5 So, A ∪ B ' = U - A ∪ B = 1 , 2 , 3 , 4 , 5 , 6 - 1 , 2 , 4 , 5 = 3 , 6 Similarly, A ' = U - A = 1 , 2 , 3 , 4 , 5 , 6 - 1 , 2 = 3 , 4 , 5 , 6 Thus, A ' ∩ B = 3 , 4 , 5 , 6 ∩ 4 , 5 = 4 , 5 Then, ( A ∪ B ) ' ∪ A ' ∩ B = 3 , 6 ∩ 4 , 5 = 3 , 4 , 5 , 6 ∴ ( A ∪ B ) ' ∪ A ' ∩ B = A ' .