HOW TO FIND COMPLEMENT OF UNION AND INTERSECTION OF A SET

Question 1 :

If U = {a, b, c, d, e, f, g, h}, A = {b, d, f, h} and B = {a, d, e, h}, find the following sets.

(i) A′ (ii) B′ (iii) A′∪B′ (iv) A′∩B′ (v) (A∪B)′

(vi) (A∩B)′ (vii) (A′)′ (viii) (B′)′

Solution :

(i)   A'  =  {a, c, e, g}

(ii)  B'  =  {b, c, f, g}

(iii)  A′∪B′   =  { a, b, c, e, g }   

(iv)  A′∩B′   =  { c, g }

(v)  (A∪B)′ 

To find (A∪B)′, first we have to find A U B

A U B  =  {a, b, d, e, f, h}

(A U B)'  =  {c, g}

(vi) (A∩B)′   =  {a, b, c, e, g }   

(vii) (A′)′

 A = {b, d, f, h}

(viii) (B′)′

B = {a, d, e, h}  

Question 2 :

Let U = {0, 1, 2, 3, 4, 5, 6, 7}, A = {1, 3, 5, 7} and B = {0, 2, 3, 5, 7}, find the following sets.

(i) A′ (ii) B′ (iii) A′∪B′ (iv)A′∩B′ ( v)(A∪B)′

(vi) (A∩B)′ (vii) (A′)′ (viii) (B′)′

Solution :

(i)   A'  =  {0, 2, 4, 6}

(ii)  B'  =  {1, 4, 6}

(iii)  A′∪B′   =  {0, 1, 2, 4, 6 }   

(iv)  A′∩B′   =  {4, 6}

(v)  (A∪B)′  

To find (A∪B)′, first we have to find A U B

A U B  =   {0, 1, 2, 3, 5, 7}

(A U B)'  =  {4, 6}

(vi) (A∩B)′   =   {0, 1, 2, 4, 6 }

(vii) (A′)′

{1, 3, 5, 7}

(viii) (B′)′

B = {0, 2, 3, 5, 7}

Question 3 :

Find the symmetric difference between the following sets.

(i) P = {2, 3, 5, 7, 11} and Q = {1, 3, 5, 11}

Solution :

P Δ Q  =  (P - Q) U (Q - P)

P - Q  =  {2, 7}

Q - P  =  {1, 11}

(P - Q) U (Q - P)  =  {1, 2, 7, 11}

Hence the value of P Δ Q is {1, 2, 7, 11}

(ii) R = {l, m, n, o, p} and S = {j, l, n, q}

Solution :

R Δ S  =  (R - S) U (S - R)

R - S  =  {m, o , p}

S - R  =  {j, q}

(R - S) U (S - R)  =  {j, m, o, p, q}

Hence the value of R Δ S is {j, m, o, p, q}.

(iii) X = {5, 6, 7} and Y = {5, 7, 9, 10}

Solution : 

X Δ Y  =  (X - Y) U (Y - X)

X - Y  =  {6}

Y - X  =  {9, 10}

(X - Y) U (Y - X)  =  {6, 9, 10}

Hence the value of X Δ Y is {6, 9, 10}.

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