RELATION & FUNCTIONS

EXERCISE # 1

1. If R is a relation from a finite set A having m elements to a finite set B having n elements, then the
number of relation from A to B is-
(1) 2mn (2) 2mn –1 (3) 2mn (4) mn

2. In the set A = {1, 2, 3, 4, 5}, a relation R is defined by R = {(x, y) | x, y   and x < y}. Then R is-
(1) Reflexive (2) Symmetric (3) Transitive (4) None of these

3. For real numbers x and y, we write x R y  x – y + 2 is an irrational number. Then the relation R is.
(1) Reflexive (2) Symmetric (3) Transitive (4) None of these

4. Let X = {1, 2, 3, 4} and Y = {1, 3, 5, 7, 9}. Which of the following is relations from X to Y-
(1) R1 = {(x, y) | y = 2 + x, x  X, y  Y}
(2) R2 = {(1, 1), (2, 1), (3, 3), (4, 3), (5, 5)}
(3) R3 = {(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)}
(4) R4 = {(1, 3), (2, 5), (2, 4), (7, 9)}

5. Let L denote the set of all straight lines in a plane. Let a relation R be defined by
 R    ⊥     L. Then R is.
(1) Reflexive (2) Symmetric (3) Transitive (4) None of these

6. Let R be a relation defined in the set of real numbers by a R b  1 + ab > 0. Then R is-
(1) Equivalence relation (2) Transitive
(3) Symmetric (4) Anti–symmetric

7. Which one of the following relations on R is equivalence relation-

(1) x R1 y  | x | = | y | (2) x R2 y  x  y
(3) x R3 y  x | y (4) x R4 y  x < y

8. Two points P and Q in a plane are related if OP = OQ, where O is a fixed point. This relation is-
(1) Reflexive but not symmetric (2) Symmetric but not transitive
(3) An equivalence relation (4) none of these

9. The relation R defined in A = {1, 2, 3} by a R b if |a2 – b2 | < 5. Which of the following is false-
(1) R = {(1, 1), (2, 2), (3, 3), (2, 1), (1, 2), (2, 3), (3, 2)}
(2) R–1 = R
(3) Domain of R= {1, 2, 3}
(4) Range of R = {5}

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10. Let a relation R in the set N of natural numbers be defined as (x, y)  R if and only if
x2 – 4xy + 3y2 = 0 for all x, y  N. The relation R is-
(1) Reflexive (2) Symmetric
(3) Transitive (4) An Equivalence relation

11. Let A = {2, 3, 4, 5} and let R = {(2, 2), (3, 3),(4, 4), (5, 5), (2, 3), (3, 2), (3, 5), (5, 3)} be a relation in A.
Then R is-
(1) Reflexive and transitive (2) Reflexive and symmetric
(3) Reflexive and antisymmetric (4) none of these

12. If A = {2, 3} and B = {1, 2}, then A×B is equal to-

(1) {(2, 1), (2, 2), (3, 1), (3, 2)} (2) {(1, 2), (1, 3), (2, 2), (2, 3)}
(3) {(2, 1), (3, 2)} (4) {(1, 2), (2, 3)}

13. Let R be a relation over the set N × N and it is defined by (a, b) R(c, d)  a + d = b + c. Then R is.
(1) Reflexive only (2) Symmetric only
(3) Transitive only (4) An equivalence relation

14. Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R(c, d) if
ad(b + c) = bc(a + d), then R is-
(1) Symmetric only (2) Reflexive only
(3) Transitive only (4) An equivalence relation

15. If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. Then
range of R is-
(1) {1, 4, 6, 9} (2) {4, 6, 9} (3) {1} (4) none of these

16. Let L be the set of all straight lines in the Euclidean plane. Two lines 1 and 2 are said to be related by
the relation R if 1 is parallel to 2 . Then the relation R is-
(1) Reflexive (2) Symmetric (3) Transitive (4) Equivalence

17. A and B are two sets having 3 and 4 elements respectively and having 2 element in common. The number
of relation which can be defined from A to B is -
(1) 25 (2) 210 – 1 (3) 212 – 1 (4) 212

18. For n, m  N, n | m means that n is a factor of m, the relation | is -

(1) reflexive and symmetric
(2) transitive and symmetric
(3) reflexive, transitive and symmetric
(4) reflexive, transitive and not symmetric

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19. Let R = {(x, y) : x, y  A, x + y = 5} where A = {1,2,3,4,5} then
(1) R is not reflexive, symmetric and not transitive
(2) R is an equivalence relation
(3) R is reflexive, symmetric but not transitive
(4) R is not reflexive, not symmetric but transitive

20. Let R be relation on a set A such that R = R–1 then R is-

(1) reflexive (2) symmetric (3) transitive (4) none of these

21. Let x, y  I and suppose that a relation R on I is defined by x R y if and only if x  y then
(1) R is partial order relation
(2) R is an equivalence relation
(3) R is reflexive and symmetric
(4) R is symmetric and transitive

22. Let R be a relation from a set A to a set B, then

(1) R = A  B (2) R = A  B (3) R  A × B (4) R  B × A

23. Given the relation R = {(1,2),(2,3)} on the set A = {1, 2, 3}, then minimum number of ordered pairs
which when added to R make it an equivalence relation
(1) 5 (2) 6 (3) 7 (4) 8

24. Let P = {(x, y) | x2 + y2 = 1, x, y  R} Then P is -

(1) reflexive (2) symmetric (3) transitive (4) anti-symmetric

25. Let X be a family of sets and R be a relation on X defined by 'A is disjoint from B'. Then R is-
(1) reflexive (2) symmetric (3) anti-symmetric (4) transitive

26. In order that a relation R defined in a non-empty set A is an equivalence relation, it is sufficient that R
(1) is reflexive (2) is symmetric
(3) is transitive (4) possesses all the above three properties

27. If R is an equivalence relation in a set A, then R–1 is -

(1) reflexive but not symmetric (2) symmetric but not transitive
(3) an equivalence relation (4) none of these

28. Let A = {p,q,r}. Which of the following is an equivalence relation in A?

(1) R1 = {(p, q), (q, r), (p, r),(p, p)} (2) R2 = {(r, q),(r, p),(r, r),(q, q)}
(3) R3 = {(p, p), (q, q), (r, r), (p, q)} (4) none of these

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 RELATION & FUNCTIONS

EXERCISE # 2 (PYQ)

1. 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 -
[AIEEE - 2004]
(1) transitive (2) not symmetric
(3) reflexive (4) a function

2. Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be relation on the set
A = {3, 6, 9, 12}. The relation is- [AIEEE - 2005]
(1) reflexive and transitive only
(2) reflexive only
(3) an equivalence relation
(4) reflexive and symmetric only

3. Let W denote the words in the English dictionary. Define the relation R by :
R = {(x, y)  W × W| the words x and y have at least one letter in common}.Then R is-
(1) reflexive, symmetric and not transitive [AIEEE - 2006]
(2) reflexive, symmetric and transitive
(3) reflexive, not symmetric and transitive
(4) not reflexive, symmetric and transitive

4. Consider the following relations :-

R = {(x, y) | x, y are real numbers and x = wy for some rational number w} ;

m p
S = { ,  | m, n, p and q are integers such that n,q  0 and qm = pn}. Then :
 n q
(1) R is an equivalence relation but S is not an equivalence relation [AIEEE - 2010]
(2) Neither R nor S is an equivalence relation
(3) S is an equivalence relation but R is not an equivalence relation
(4) R and S both are equivalence relations

5. Let R be the set of real numbers. [AIEEE - 2011]

Statement-1 : A = {(x, y)  R × R : y – x is an integer} is an equivalence relation on R.
Statement-2: B = {(x, y)  R × R : x = y for some rational number } is an equivalence relation on R.
(1) Statement-1 is true, Statement-2 is false.
(2) Statement-1 is false, Statement-2 is true
(3) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
(4) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

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 RELATION & FUNCTIONS
6. Consider the following two binary relations on the set
A = {a, b, c} :
R1 = {(c, a), (b, b), (a, c), (c, c), (b, c) (a, a)} and
R2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}.
Then
(1) R1 is not symmetric but it is transitive (2) both R1 and R2 are transitive
(3) R2 is symmetric but is not transitive (4) both R1 and R2 are not symmetric
[JEE Mains Online-2018]

7. Let N denote the set of all natural numbers. Define two binary relations on N as
R1 = {(x, y)  N × N : 2x + y = 10} and
R2 = {(x, y)  N × N : x + 2y = 10}. Then :
(1) Both R1 and R2 are symmetric relations (2) Range of R1 is {2, 4, 8}
(3) Both R1 and R2 are transitive relations (4) Range of R2 is {1, 2, 3, 4}
[JEE Mains Online-2018]

8. Let A = 2,3,4,5,.....,30 and ' ' be an equivalence relation on A  A, defined by ( a,b) ( c,d ) , if an
only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered
pair (4, 3) is equal to: [JEE Mains-2021]
(1) 5 (2) 6 (3) 8 (4) 7

9. Define a relation R over a class of n × n real matrices A and B as “ARB iff there exists a non-singular
matrix P such that P AP–1 = B". Then which of the following is true? [JEE Mains-2021]
(1) R is symmetric, transitive but not reflexive
(2) R is reflexive, symmetric but not transitive
(3) R is an equivalence relation
(4) R is reflexive, transitive but not symmetric

10. Which of the following is not correct for relation R on the set of real numbers? [JEE Mains-2021]
(1) ( x, y )  R  0 | x | – | y | 1 is neither transitive nor symmetric
(2) ( x, y )  R  0 | x – y | 1 is symmetric and transitive
(3) ( x, y )  R | x | – | y |  1 is reflexive but not symmetric
(4) ( x, y )  R | x – y |  1 is reflexive and symmetric

11. Let be the set of all integers,


A = ( x, y)   : ( x – 2) + y2  4
2

B = ( x, y )   : x 2 + y2  4 and


C = ( x, y)   : ( x – 2) + ( y – 2)  4
2 2

If the total number of relation from A  B to A  C is 2p , then the value of p is : [JEE Mains-2021]
(1) 16 (2) 25 (3) 49 (4) 9

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12. Let N be the set of natural numbers and a relation R on N be defined by
R = ( x, y )  N  N : x3 – 3x 2 y – xy2 + 3y3 = 0 Then the relation R is: [JEE Mains-2021]
(1) symmetric but neither reflexive nor transitive (2) reflexive but neither symmetric nor transitive
(3) reflexive and symmetric, but not transitive (4) an equivalence relation

13. 
Let R1 and R 2 be relations on the set 1,2,....,50 such that R1 = ( p,pn ) : p is a prime and n  0 is an


integer} and R 2 = ( p,pn ) : p is a prime and n=0 or 1}. Then, the number of elements in R1 – R2 is____.
[JEE Mains-2022]

14. Let R1 = ( a,b)  N  N: a – b  13 and R2 = ( a,b)  N  N: a – b  13. Then on N:

[JEE Mains-2022]
(1) Both R1 and R 2 are equivalence relations
(2) Neither R 1 nor R 2 is an equivalence relation
(3) R1 is an equivalence relation but R2 is not
(4) R2 is an equivalence relation but R1 is not

15. Let a set A = A1  A2 .....  A k' where Ai  A j =  for i  j1  i, j  k. Define the relation r from A to

A by R = ( x, y ) : y  Ai if and only if x  Ai ,1  i  k . Then, R is: [JEE Mains-2022]

(1) reflexive, symmetric but not transitive (2) reflexive transitive but not symmetric
(3) reflexive but not symmetric and transitive (4) an equivalence relation

16. The probability that a relation R from x, y to x, y is both symmetric and transitive, is equal to:
[JEE Mains-2022]
5 9 11 13
(1) (2) (3) (4)
16 16 16 16

17. Let R1 and R2 be two relations defined on by aR1b  ab  0 and aR2b  a  b, then
[JEE Mains-2022]
(1) R1 is an equivalence relation but not R2 (2) R2 is an equivalence relation but not R1
(3) Both R1 and R2 are equivalence relation (4) neither R1 nor R2 is an equivalence relation

18. For  N, consider a relation R on N given by R = ( x, y ) :3x + y is a multiple of 7}. The relation R is
an equivalence relation if and only if: [JEE Mains-2022]
(1)  = 14 (2)  is a multiple of 4
(3) 4 is the remainder when  is divided by 10 (4) 4 is the remainder when  is divided by 7

19. Let R be a relation from the set {1, 2, 3, ….., 60} to itself such that R = ( a,b) : b = pq, where p,q  3
are prime number}. Then, the number of elements in R is [JEE Mains-2022]
(1) 600 (2) 660 (3) 540 (4) 720

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Answer Key

EXERCISE # 1
1. (1) 2. (3) 3. (1) 4. (1) 5. (2) 6. (3) 7. (1)
8. (3) 9. (4) 10. (1) 11. (2) 12. (1) 13. (4) 14. (4)
15. (3) 16. (4) 17. (4) 18. (4) 19. (1) 20. (2) 21. (1)
22. (3) 23. (3) 24. (2) 25. (2) 26. (4) 27. (3) 28. (4)

EXERCISE # 2 (PYQ)
1. (2) 2. (1) 3. (1) 4. (3) 5. (1) 6. (3) 7. (4)
8. (4) 9. (3) 10. (2) 11. (2) 12. (2) 13. (8) 14. (2)
15. (4) 16. (1) 17. (4) 18. (4) 19. (2)

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