## COMMUTATIVE PROPERTY OF SET

For any two two sets, the following statements are true.

(i)  Set union is commutative

A u B = B u A

(i)  Set intersection is commutative

A n B = B n A

Example 1 :

A = {-10, 0, 1, 9, 2, 4, 5} and B = {- 1,- 2, 5, 6, 2, 3, 4}, for the sets A and B, verify that

(i) Set union is commutative. Also verify it by using Venn diagram.

(ii) Set intersection is commutative. Also verify it by using Venn diagram.

Solution :

Commutative property :

(i) A u B = B u A

(ii) A n B = B n A

(i) A = {-10, 0, 1, 9, 2, 4, 5} and B = {- 1, -2, 5, 6, 2, 3, 4}

A u B = {-10, 0, 1, 9, 2, 4, 5} u {-1,-2, 5, 6, 2, 3, 4}

= {-10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9} ----(1)

B u A = {-1, -2, 5, 6, 2, 3, 4} u {-10, 0, 1, 9, 2, 4, 5}

=  {-10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9} ----(2)

(1) = (2)

Verifying in Venn diagram :

(ii) A = {-10, 0, 1, 9, 2, 4, 5} and B = {-1,-2, 5, 6, 2, 3, 4}

A n B = {-10, 0, 1, 9, 2, 4, 5} n {-1,-2, 5, 6, 2, 3, 4}

= { 2, 4, 5 } ----(3)

B n A {-1,-2, 5, 6, 2, 3, 4} n {-10, 0, 1, 9, 2, 4, 5}

= { 2, 4, 5 } ----(4)

(3) = (4)

Verifying in Venn diagram :

Example 2 :

A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6}, for the sets A and B, verify that

(i) set union is commutative.

(ii) set intersection is commutative.

Solution :

Commutative property :

(i) A u B = B u A

(ii) A n B = B n A

(i) A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6}

A u B = {1, 2, 3, 4, 5} u {3, 4, 5, 6}

= {1, 2, 3, 4, 5, 6} ----(5)

B u A = {3, 4, 5, 6} u {1, 2, 3,  4, 5}

= {1, 2, 3, 4, 5, 6} ----(6)

(5) = (6)

(i) A = {1, 2, 3,  4, 5} and B = {3, 4, 5, 6}

A n B = {1, 2, 3, 4, 5} n {3, 4, 5, 6}

= {3, 4, 5} ----(7)

B n A =  {3, 4, 5, 6} U {1, 2, 3,  4, 5}

= {3, 4, 5} ----(8)

(7) = (8)

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