PRACTICE PROBLEMS ON PROPERTIES OF SET THEORY

Commutative Property :

For any two sets A and B

(i) A U B = B U A

(ii) A n B = B n A

Associative Property :

For any three sets A, B and C

(i) A U (B U C) = (A U B) U C

(ii)A n (B n C) = (A n B) n C

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)  =  { } ------(1)

(A n B)  =  { 5 }

 (A n B) n C  =  {   ------(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.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. SAT Math Resources (Videos, Concepts, Worksheets and More)

    Oct 01, 24 12:04 PM

    SAT Math Resources (Videos, Concepts, Worksheets and More)

    Read More

  2. Digital SAT Math Problems and Solutions (Part - 47)

    Oct 01, 24 11:49 AM

    Digital SAT Math Problems and Solutions (Part - 47)

    Read More

  3. Digital SAT Math Problems and Solutions (Part - 46)

    Sep 30, 24 11:38 AM

    digitalsatmath41.png
    Digital SAT Math Problems and Solutions (Part - 46)

    Read More