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 at some example problems based on above properties.

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)

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 :

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}, 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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