Question 1 :
If A = {x : x ∊ Z,−2 < x ≤ 4}, B={x : x ∈ W, x ≤ 5}, C = {−4,−1, 0,2, 3, 4}, then verify A U (B n C) = (A U B) n (A U C).
Solution :
A = {x : x ∊ Z,−2 < x ≤ 4}
A = {-1, 0, 1, 2, 3, 4}
B = {x : x ∈ W, x ≤ 5}
B = {0, 1, 2, 3, 4, 5}
C = {−4,−1, 0, 2, 3, 4}
A U (B n C) = (A U B) n (A U C)
(B n C) = {0, 2, 3, 4}
A U (B n C) = {-1, 0, 1, 2, 3, 4} ----(1)
(A U B) = {-1, 0, 1, 2, 3, 4, 5}
(A U C) = {-4, -1, 0, 1, 2, 3, 4}
(A U B) n (A U C) = {-1, 0, 1, 2, 3, 4}----(2)
(1) = (2)
Question 2 :
Verify A U (B n C) = (A U B) n (A U C) using Venn diagrams
Solution :
Hence proved.
Question 3 :
If A = {b, c, e, g, h} , B = {a, c, d, g, i} and C = {a, d, e, g, h} then show that A − (B n C) = (A − B) U (A−C) .
Solution :
(B n C) = {a, d, g}
A - (B n C) = {b, c, e, h} ----(1)
(A − B) = {b, e, h}
(A − C) = {b, c}
(A − B) U (A−C) = {b, c, e, h} ----(2)
(1) = (2)
Hence proved
Question 4 :
If A = {x : x = 6n, n ∈ W and n < 6}, B = {x : x = 2n, n ∈ N and 2 < n ≤ 9} and C = {x : x = 3n, n ∈ N and 4 ≤ n < 10}, then show that A−(B n C) = (A − B) U (A − C)
Solution :
A = {x : x = 6n, n ∈ W and n < 6}
A = {0, 6, 12, 18, 24, 30}
B = {x : x = 2n, n ∈ N and 2 < n ≤ 9}
B = {6, 8, 10, 12, 14, 16, 18}
and C = {x : x = 3n, n ∈ N and 4 ≤ n < 10}
C = {12, 15, 18, 21, 24, 27}
(B n C) = {12, 18}
A−(B n C) = {0, 6, 24, 30} ----(1)
(A − B) = {0, 24, 30}
(A − C) = {0, 6, 30}
(A − B) U (A − C) = {0, 6, 24, 30} ----(2)
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