DIFFERENCE OF TWO SETS PRACTICE QUESTIONS

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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