The following problems are under difference of sets.
Question 1 :
Given that U = {3, 7, 9, 11, 15, 17, 18}, M = { 3, 7, 9, 11} and N = { 7, 11, 15, 17},
find (i) M-N (ii) N-M (iii) N'-M (iv) M'-N
(v) M∩(M-N) (vi) N∪(N-M) (vii) n(M-N)
Question 2 :
If A = {3, 6, 9, 12, 15, 18}, B = {4, 8, 12, 16, 20},
C = {2, 4, 6, 8, 10, 12} and D = {5, 10, 15, 20, 25}, find
(i) A-B (ii) B-C (iii) C-D
(iv) D-A (v) n(A-C) (vi) n(B-A)
Question 3 :
Let U = {x : x is a positive integer less than 50},
A = {x: x is divisible by 4} and B = {x : x leaves remainder 2 when divided by 14}
(i) List the elements of U, A and B
(ii) Find (i) A∪B (ii) A∩B (iii) (A∪B)' (iv) n( A∩B)
(v) A-B and (vi) B-A
Question 4 :
Find the symmetric difference between the following sets.
(i) X = {a, d, f, g,h} and Y = {b, e, g, h, k}
(ii) P = {x: 3<x <9, x∈ℕ} and Q = { x: x<5, x∈W}
(iii) A = {-3, -2,0, 2, 3, 5} and B = {-4,-3,-1, 0, 2, 3}
(1) Solution :
U = {3, 7, 9, 11, 15, 17, 18}
M = {3, 7, 9, 11} and M' = {15, 17, 18}
N = {7, 11, 15, 17} and N' = {3, 9, 18}
(i) M-N = {3, 9}
(ii) N-M = {15, 17}
(iii) N'-M = {18}
(iv) M'-N = {18}
(v) M∩(M-N) = {3, 7, 9, 11} ∩ {3, 9}
M∩(M-N) = {3, 9}
(vi) N∪(N-M) = { 7, 11, 15, 17} ∪ {15,17}
N∪(N-M) = {7, 11, 15, 17}
(vii) n(M-N) = 2
(2) Solution :
Given A = {3, 6, 9, 12, 15, 18}, B = {4, 8, 12, 16, 20},
C = {2, 4, 6, 8, 10, 12} and D = {5, 10, 15, 20, 25}
(i) A-B = (Elements only in A not in B)
A-B = {3, 6, 9 15, 18}
(ii) B-C = {16, 20}
(iii) C-D = {2, 4, 6, 8,12}
(iv) D-A = {5, 10, 20, 25}
(v) (A-C) = {3, 9, 15, 18} ==> n(A-C) = 4
(vi) (B-A) = {4, 8, 16, 20} ==> n(B-A) = 4
(3) Solution :
First let us write the given sets.
(i) U = {1, 2, 3, 4, 5, 6, 7.......49}
A = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}
B = {16, 30,44}
(ii) A∪B
= {4, 8, 12, 16, 20, 24, 28, 30, 32, 36, 40, 44, 48}
(iii) A∩B = {16, 44}
(iii) (A∪B)'
= {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49}
(iv) n(A∩B) = 2
(v) A-B = {4, 8, 12, 20, 24, 28, 32, 36, 40, 48}
B-A = {30}
(4) Solution :
(i) Symmetric difference between X and Y is
X = {a, d, f, g, h} and Y = {b, e, g, h, k}
X∆Y = (X-Y) ∪ (Y-X)
(X-Y) = {a, d, f, g, h} - {b, e, g, h, k}
X-Y = {a, d, f} ---(1)
Y-X = {b, e, k} ---(2)
(X-Y) ∪ (Y-X) = {a, d, f} ∪ {b, e, k}
(X-Y) ∪ (Y-X) = {a, b, d, e, f, k}
(ii) P = {x: 3<x <9, x∈ℕ} and Q = { x: x<5, x∈W}
P = {4, 5, 6, 7, 8}, Q = {0, 1, 2, 3, 4}
Symmetric difference between P and Q is
P∆Q = (P-Q)∪(Q-P)
P-Q = {4, 5, 6,7,8}- {0, 1, 2, 3, 4}
Q-P = {0, 1, 2, 3, 4}-{4, 5, 6, 7, 8}
(P-Q)∪(Q-P) = {5, 6, 7, 8} ∪ {0, 1, 2, 3}
(P-Q)∪(Q-P) = {0, 1, 2, 3, 5, 6, 7, 8}
(iii) A = {-3, -2, 0, 2, 3, 5} and B = {-4, -3, -1, 0, 2, 3}
Symmetric difference of A and B is
A∆B = (A-B)∪(B-A)
(A-B) = {-3, -2, 0, 2, 3, 5} - {-4, -3, -1, 0, 2, 3}
(A-B) = {-2, 5}
(B-A) = {-4, -3, -1, 0, 2, 3} - {-3, -2, 0, 2, 3, 5}
(B-A) = {-4, -1}
(A-B)∪(B-A) = {-2, 5} ∪ {-4, -1}
(A-B)∪(B-A) = {-4, -2, -1, 5}
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