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RELATIONS AND FUNCTIONS

1. Both the empty relation and universal relation is

(A)Empty relation (B)Universal relation (C) Trivial relation (D) Equivalence relation
Ans (C)
2. A relation R in a set A, if “each elements of A is related to every element of A ” then R is called
(A) Empty relation (B)Universal relation (C) Trivial relation (D) Equivalence relation
Ans (B)
3. A relation R in the set A is called a reflexive relation if,

(A)  a,a   R, for some a  A (B) If  a,b   R then  b,a   R for a,b  A

(C) If  a,b  , b,c   R then  a,c   R for a,b,c  A (D)  a,a   R, for every a  A

Ans: (D)

4. Let R be the relation in the set N given by R = {(a, b): a = b  2, b > 6}, then

(A) (2, 4)  R (B) (3, 8)  R (C) (6, 8)  R (D) (8, 7)  R
Ans (C)
5. If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is
(A) reflexive (B) transitive (C) symmetric (D) an equivalence relation
Ans (B)
6. Let P = {(x, y) |x + y = 1, x, y  R}. Then, P is
2 2


(A) reflexive (B) symmetric (C) transitive (D) anti-symmetric
Ans (B)
The relation is neither reflexive nor transitive but it is symmetric, because
x2 + y2 = 1 and y2 + x2 = 1
7. Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is
(A) not symmetric (B) transitive (C) a function (D) reflexive
Ans (A)
(2, 3)  R but (3, 2)  R
 R is not symmetric.

8. Consider the non-empty set consisting of children in a family and a relation R defined as aRb, if a is
brother of b. then R is
(A) Symmetric but not transitive (B) Transitive but not symmetric

Prepared by: Dr. H. T. PRAKASHA (H T P) BASE EDUCATION (RV PU COLLEGE)

, (C) Neither symmetric nor Transitive (D) both symmetric and transitive
Ans (B)

9. Let L denote the set of all straight lines in a plane. Let relation R be defined by lRm if and only if l is
perpendicular to m , for all l , m  L . Then R is

(A) reflexive (B) transitive (C) symmetric (D) an equivalence relation

Ans (C)
10. Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and
symmetric but not transitive is
(A) 1 (B) 2 (C) 3 (D) 4
Ans (A)
We get only one relation R = (1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 1), (3, 1)

11. If R1 and R2 are two equivalence relations on a non-empty set A, then R1  R2 is not
(A) reflexive (B) symmetric (C) transitive (D) an equivalence relation
Ans (C)
Let A = {1, 2, 3}
Consider the example R1 = {(1, 1) (2, 2) (3, 3) (1, 2) (2, 1)}
R2 = {(1, 1) (2, 2) (3, 3) (2, 3) (3, 2)}
Here R1 and R2 are equivalence.
But R1  R2 is not transitive. For R1  R2 = {(1, 1) (2, 2) (3, 3) (1, 2) (2, 1) (2, 3) (3, 2)}
(1, 2)  R1  R2, (2, 3)  R1  R2 but (1, 3)  R1  R2

12. Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is
(A) 1 (B) 2 (C) 3 (D) 4
Ans (B)
R1 = (1, 1), (2, 2), (3, 3), (1, 2), (2, 1)
R2 = (1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2) are the equivalence relations.

13. The number of equivalence relations defined in the set S = {a, b, c} is
(A) 5 (B) 3! (C) 22 (D) 33
Ans (A)

14. Which of the following relation in the set 1, 2,3 is symmetric and transitive but not reflexive?

A) 1, 2  , 2, 2  ,1,1 (B) 1,2 , 2,1 (C)  2,3 (D) 1,2 , 2,1 , 2,2 ,1,1
Ans (D)
15. Let R be a relation on the set N of Natural numbers defined by nRm if “n divides m” then R is

Prepared by: Dr. H. T. PRAKASHA (H T P) BASE EDUCATION (RV PU COLLEGE)