ASSOCIATIVE PROPERTY SETS

Associative property of sets :

Here we are going to see the associative property used in sets.

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

(i)  Set union is associative

A U (B U C) = (A U B) U C

(i)  Set intersection is associative

A n (B n C) = (A n B) n C

Let us look into some example problems based on above properties.

Associative property of set - Examples

Example 1 :

Given, A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8}, show that

(i) A U (B U C) = (A U B) U C. (ii) Verify (i) using Venn diagram.

Solution :

Now, B U C = {3, 4, 5, 6} U {5, 6, 7, 8} = {3, 4, 5, 6, 7, 8}

A U (B U C)  =  {1, 2, 3, 4, 5} U { 3, 4, 5, 6, 7, 8}

=  {1, 2, 3, 4, 5, 6, 7, 8}  -----  (1)

Now, (A U B) = {1, 2, 3, 4, 5} U {3, 4, 5, 6} = {1,2,3,4,5,6}

(A U B) U C = {1,2,3,4,5,6} U {5,6,7,8}

=  {1, 2, 3, 4, 5, 6, 7, 8}   -----  (2)

(1)  =  (2)

Example 2 :

Let A = {a,b,c,d}, B = {a,c,e} and C = {a,e}.

(i) Show that A n (B n C) = (A n B) n C. (ii) Verify (i) using Venn diagram

Solution :

Now, B n C = {a,c,e} n {a, e} = {a, e}

A n (B n C)  =  {a, b, c ,d} U {a, e}

=  { a }  -----  (1)

(A n B)  =  {a,b,c,d} n {a,c,e}

=  {a, c}

(A n B) n C  =  {a, c} n {a,e}

  =  {a}  -----  (2)

(1)  =  (2)

Example 3 :

For A = {x ; x is a prime factor of 42}, B = {x | 5  x  12, x  N} and C = {1, 4, 5, 6}, verify A U (B U C) = (A U B) U C

Solution :

A = {2, 3, 7}, B = {5, 6, 7, 8, 9, 10, 11, 12} and C = {1, 4, 5, 6}

(B U C)  =  {5, 6, 7, 8, 9, 10, 11, 12} U {1, 4, 5, 6}

  =  {1, 2, 4, 6, 8, 9, 10, 11, 12}

A U (B U C)  =  {2, 3, 7} U {1, 2, 4, 6, 8, 9, 10, 11, 12}

 =  {1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12}  ----(1)

(A U B)  =  {2, 3, 7} U {5, 6, 7, 8, 9, 10, 11, 12}

=  {2, 3, 5, 6, 7, 8, 9, 10, 11, 12}

(A U B) U C  =  {2, 3, 5, 6, 7, 8, 9, 10, 11, 12} U {1, 4, 5, 6}

 =  {1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12}  ----(2)

(1)  =  (2)

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