PRACTICE PROBLEMS ON PROPERTIES OF SET THEORY

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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