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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