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)

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. SAT Math Resources (Videos, Concepts, Worksheets and More)

    Jul 11, 24 05:20 PM

    SAT Math Resources (Videos, Concepts, Worksheets and More)

    Read More

  2. SAT Math Videos (Part - 21)

    Jul 11, 24 05:14 PM

    satmath18.png
    SAT Math Videos (Part - 21)

    Read More

  3. Best Way to Learn Mathematics

    Jul 11, 24 11:59 AM

    Best Way to Learn Mathematics

    Read More