Commutative Property :
For any two sets A and B
(i) A U B = B U A
(ii) A n B = B n A
Associative Property :
For any three sets A, B and C
(i) A U (B U C) = (A U B) U C
(ii)A n (B n C) = (A n B) n C
Question 1 :
If P = {1,2,5,7,9}, Q = {2, 3,5,9,11}, R = {3, 4,5,7,9} and S = {2, 3, 4,5, 8}, then find (i) (P U Q) U R (ii) (P n Q) n S (iii) (Q n S) n R
Solution :
P = {1, 2, 5, 7, 9}
Q = {2, 3, 5, 9, 11}
R = {3, 4, 5, 7, 9}
S = {2, 3, 4, 5, 8}
(i) (P U Q) U R
(P U Q) = {1, 2, 3, 5, 7, 9, 11}
(P U Q) U R = {1, 2, 3, 4, 5, 7, 9, 11} -------(1)
(ii) (P n Q) n S
P n Q = {2, 5, 9}
(P n Q) n S = {2, 5} ------(2)
(iii) (Q n S) n R
Q n S = {2, 3, 5}
(Q n S) n R = {3, 5} ------(3)
Question 2 :
Test for the commutative property of union and intersection of the sets
P = { x : x is a real number between 2 and 7} and
Q = { x : x is an irrational numbers between 2 and 7}
Solution :
P = {3, 4, 5, 6}
Q = {2.01......, .............., 6.990,..............}
P U Q = Q U P
P n Q = Q n P
Question 3 :
If A = {p, q, r, s}, B = {m, n, q, s, t} and C = {m, n, p, q, s}, then verify the associative property of union of sets.
Solution :
Associative property for union :
A U (B U C) = (AU B) U C
B U C = {m, n, p, q, s, t}
A U (B U C) = {m, n, p, q, r, s, t} ------(1)
(AU B) = {m, n, p, q, r, s, t}
(AU B) U C = {m, n, p, q, r, s, t} ------(2)
(1) = (2)
Hence proved.
Question 4 :
Verify the associative property of intersection of sets for A = {−11, √2, √5, 7}, B = {√3, √5, 6, 13} and C = {√2, √3, √5, 9}.
Solution :
Associative property for intersection :
A n (B n C) = (A n B) n C
B n C = {√3, √5}
A n (B n C) = { √5 } ------(1)
(A n B) = { √5 }
(A n B) n C = { √5 } ------(2)
(1) = (2)
Hence proved.
Question 5 :
If A = {x : x = 2n, n ∈ W and n < 4}, B = {x : x = 2n, n ∈ N and n ≤ 4} and C = {0, 1, 2, 5, 6}, then verify the associative property of intersection of sets.
Solution :
A = {x : x = 2n, n ∈ W and n < 4}
A = {1, 4, 8}
B = {x : x = 2n, n ∈ N and n ≤ 4}
B = { 2, 4, 6, 8 }
C = {0, 1, 2, 5, 6}
Associative property for intersection :
A n (B n C) = (A n B) n C
(B n C) = {2, 6}
A n (B n C) = { } ---(1)
(A n B) = {4, 8}
(A n B) n C = { } ---(2)
Hence proved.
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