JEE Main - Relations and types of relations Question Bank with Options

1. Which of the following relations on the set of natural numbers N is NOT an equivalence relation?

A) x R y if x divides y

B) x R y if x = y

C) x R y if x + y is even

D) x R y if x - y is divisible by 3

2. Let A = {1, 2, 3, 4} and R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (2,3), (3,2)}. Is R transitive?

A) No, because (1,2) ∈ R and (2,3) ∈ R, but (1,3) ∉ R

B) Yes, because it is reflexive and symmetric

C) Yes, because all pairs satisfy the condition.

D) No, because (3,1) is not in R

3. Let A = {1, 2, 3, 4} and R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1)}. Is R transitive?

A) No, because (1,2) ∈ R and (2,1) ∈ R, but (1,1) ∈ R. This condition is met. However, if we consider (2,1) ∈ R and (1,2) ∈ R, then (2,2) ∈ R. Let's recheck.

B) Yes, because it is reflexive and symmetric

C) No, because (1,3) is not in R

D) Yes, all pairs satisfy the condition.

4. If R is an equivalence relation on A, it partitions the set A into disjoint subsets called:

A) Equivalence classes

B) Subsets

C) Partitions

D) Disjoint sets

5. Let R be a relation on a set A. Which of the following is NOT a necessary condition for R to be an equivalence relation?

A) R must be reflexive

B) R must be symmetric

C) R must be transitive

D) R must contain at least one non-diagonal pair (a,b) where a != b

6. Since the relation 'x - y is divisible by 5' on integers is reflexive, symmetric, and transitive, it is an:

A) Equivalence relation

B) Universal relation

C) Empty relation

D) Identity relation

7. Consider the relation R on the set Z of integers defined by x R y if x - y is divisible by 5. Is R transitive?

A) Yes, because if x - y = 5k and y - z = 5m (k, m integers), then x - z = (x - y) + (y - z) = 5k + 5m = 5(k + m), which is divisible by 5

B) No, because if x - y is divisible by 5, then y - x is not

C) Yes, because it is reflexive and symmetric

D) No, because it is not symmetric

8. Consider the relation R on the set Z of integers defined by x R y if x - y is divisible by 5. Is R symmetric?

A) Yes, because if x - y = 5k for some integer k, then y - x = -5k = 5(-k), which is also divisible by 5

B) No, because x - y being divisible by 5 does not mean y - x is

C) Yes, because it is reflexive

D) No, because it is not transitive

9. Consider the relation R on the set Z of integers defined by x R y if x - y is divisible by 5. Is R reflexive?

A) Yes, because x - x = 0, which is divisible by 5

B) No, because x - y being divisible by 5 does not mean y - x is

C) Yes, because if x - y is divisible by 5 and y - z is divisible by 5, then x - z is divisible by 5

D) No, because it is not symmetric

10. Let R be a relation on set A. If R is symmetric and transitive, does it imply R is reflexive?

A) No, not necessarily. For example, R = {(1,2), (2,1)} on A = {1, 2, 3}.

B) Yes, always.

C) Yes, if A has only one element.

D) No, unless R is also the universal relation.

11. What property is shared by both the empty relation and the universal relation on a non-empty set?

A) Transitivity

B) Symmetry

C) Reflexivity

D) Irreflexivity

12. Let A = {1, 2, 3} and R = {(1,2), (2,3), (3,1)}. Is R transitive?

A) No, because (1,2) ∈ R and (2,3) ∈ R, but (1,3) ∉ R

B) Yes, because it is a cycle

C) Yes, because it is symmetric

D) No, because it is not reflexive

13. The relation 'is perpendicular to' on the set of all lines in a plane is:

A) Symmetric but not reflexive or transitive

B) Reflexive, symmetric, and transitive

C) Irreflexive and transitive

D) Symmetric and transitive but not reflexive

14. Consider the relation R on the set of all lines in a plane defined by L1 R L2 if L1 is parallel to L2. Is this relation transitive?

A) Yes, because if L1 is parallel to L2 and L2 is parallel to L3, then L1 is parallel to L3

B) No, because L1, L2, L3 must be distinct

C) Yes, because it is reflexive and symmetric

D) No, because it is not symmetric

15. Consider the relation R on the set of all lines in a plane defined by L1 R L2 if L1 is parallel to L2. Is this relation symmetric?

A) Yes, because if L1 is parallel to L2, then L2 is parallel to L1

B) No, because if L1 || L2, then L1 and L2 are distinct

C) Yes, because it is reflexive

D) No, because it is not transitive

16. Consider the relation R on the set of all lines in a plane defined by L1 R L2 if L1 is parallel to L2. Is this relation reflexive?

A) Yes, because every line is parallel to itself

B) No, because a line cannot be parallel to another distinct line

C) Yes, because if L1 || L2, then L2 || L1

D) No, because not all lines are parallel to each other

17. Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,3)}. Is R transitive on A?

A) No, because (1,2) ∈ R and (2,3) ∈ R, but (1,3) ∉ R

B) Yes, because it is reflexive

C) Yes, because it is symmetric

D) No, because (2,1) is not in R

18. Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Is R transitive on A?

A) No, because (1,2) ∈ R and (2,1) ∈ R, but (1,1) ∈ R. Let's check other combinations. (1,2) ∈ R and (2,2) ∈ R, implies (1,2) ∈ R. (2,1) ∈ R and (1,1) ∈ R, implies (2,1) ∈ R. The issue arises from (1,2) and (2,1).

B) Yes, because it is reflexive and symmetric

C) No, because (3,3) is not involved in transitivity

D) Yes, because all pairs satisfy the condition.

19. Let A = {1, 2} and R = {(1,1), (1,2), (2,1)}. Is R transitive on A?

A) No, because (1,2) ∈ R and (2,1) ∈ R, but (1,1) is present. However, consider (1,2) and (2,1). We have (1,1) ∈ R. Now consider (2,1) and (1,1). We need (2,1) ∈ R, which is true. The issue is if we have (a,b) and (b,c) implies (a,c). Here, (1,2) ∈ R and (2,1) ∈ R, so we must have (1,1) ∈ R, which is true. Let's re-evaluate.

B) No, because (2,2) ∉ R

C) Yes, because it is reflexive and symmetric

D) No, because (1,2) ∈ R and (2,1) ∈ R, but (1,1) ∈ R is not sufficient for transitivity in all cases.

20. Let A = {1, 2} and R = {(1,1), (1,2), (2,1)}. Is R symmetric on A?

A) Yes, because (1,2) ∈ R implies (2,1) ∈ R and (1,1) ∈ R implies (1,1) ∈ R

B) No, because (2,2) ∉ R

C) Yes, because it is reflexive

D) No, because it is not transitive

21. Let A = {1, 2} and R = {(1,1), (1,2), (2,1)}. Is R reflexive on A?

A) No, because (2,2) ∉ R

B) Yes, because (1,1) ∈ R

C) Yes, because it is symmetric

D) No, because it is not transitive

22. If a relation R on set A is not transitive, what does it mean?

A) There exist elements a, b, c ∈ A such that (a, b) ∈ R and (b, c) ∈ R but (a, c) ∉ R

B) For all elements a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∉ R

C) There exists at least one element a ∈ A such that (a, a) ∉ R

D) There exists at least one pair (a, b) ∈ R such that (b, a) ∉ R

23. If a relation R on set A is not symmetric, what does it mean?

A) There exists at least one pair (a, b) ∈ R such that (b, a) ∉ R

B) For all pairs (a, b) ∈ R, (b, a) ∉ R

C) There exists at least one element a ∈ A such that (a, a) ∉ R

D) There exist a, b, c ∈ A such that (a, b) ∈ R and (b, c) ∈ R but (a, c) ∉ R

24. If a relation R on set A is not reflexive, what does it mean?

A) There exists at least one element a ∈ A such that (a, a) ∉ R

B) For all elements a ∈ A, (a, a) ∉ R

C) There exists at least one pair (a, b) ∈ R such that (b, a) ∉ R

D) There exist a, b, c ∈ A such that (a, b) ∈ R and (b, c) ∈ R but (a, c) ∉ R

25. Let A = {a, b, c}. Which relation is the identity relation on A?

A) {(a,a), (b,b), (c,c)}

B) {(a,b), (b,a), (c,c)}

C) {(a,a), (b,b)}

D) A x A

26. What is the empty relation on a set A?

A) The relation ∅

B) The relation A x A

C) The identity relation I_A

D) A relation that is irreflexive only

27. What is the universal relation on a set A?

A) The relation A x A

B) The empty relation ∅

C) The identity relation I_A

D) A relation that is reflexive only

28. Which type of relation is reflexive, symmetric, and transitive?

A) Equivalence relation

B) Universal relation

C) Empty relation

D) Identity relation

29. Consider the relation R on the set of integers defined by x R y if x > y. Is this relation transitive?

A) Yes, because if x > y and y > z, then x > z

B) No, because if x > y and y > z, then x = z is not guaranteed

C) Yes, because it is irreflexive

D) No, because it is not symmetric

30. Consider the relation R on the set of integers defined by x R y if x > y. Is this relation symmetric?

A) No, because if x > y, then y > x is false (e.g., 5 > 3 but 3 is not > 5)

B) Yes, because if x > y, then y < x

C) Yes, because it is transitive

D) No, because it is not reflexive

31. Consider the relation R on the set of integers defined by x R y if x > y. Is this relation irreflexive?

A) Yes, because x > x is never true for any integer x

B) No, because x > y does not imply y > x

C) Yes, because if x > y and y > z, then x > z

D) No, because it is not symmetric

32. What is an irreflexive relation on a set A?

A) A relation where for all a ∈ A, (a, a) ∉ R

B) A relation where for all a, b ∈ A, if (a, b) ∈ R, then (b, a) ∉ R

C) A relation where for all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∉ R

D) A relation where (a, a) ∈ R for at least one a ∈ A

33. Let A = {1, 2, 3} and R = {(1,2), (2,1)}. Is R transitive on A?

A) No, because (1,1) is not in R

B) Yes, because there are no elements a, b, c such that (a, b) ∈ R and (b, c) ∈ R but (a, c) ∉ R

C) Yes, because it is reflexive

D) No, because it is not symmetric

34. Let A = {1, 2, 3} and R = {(1,2), (2,1)}. Is R symmetric on A?

A) No, because (1,1), (2,2), (3,3) are not in R

B) Yes, because if (1,2) ∈ R, then (2,1) ∈ R

C) Yes, because it is reflexive

D) No, because it is not transitive

35. Let A = {1, 2, 3} and R = {(1,2), (2,1)}. Is R reflexive on A?

A) No, because (1,1), (2,2), (3,3) are not in R

B) Yes, because if (1,2) ∈ R, then (2,1) ∈ R

C) Yes, because it is symmetric

D) No, because it is not transitive

36. Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3)}. Which type of relation is R on A?

A) Reflexive, Symmetric, and Transitive

B) Only Reflexive

C) Only Symmetric

D) Only Transitive

37. Consider the relation R on the set of real numbers defined by x R y if x ≤ y. Is this relation transitive?

A) Yes, because if x ≤ y and y ≤ z, then x ≤ z

B) No, because if x ≤ y and y ≤ z, then x = z is not guaranteed

C) Yes, because it is reflexive and symmetric

D) No, because it is not symmetric

38. Consider the relation R on the set of real numbers defined by x R y if x ≤ y. Is this relation symmetric?

A) No, because x ≤ y does not imply y ≤ x (e.g., 2 ≤ 3 but 3 is not ≤ 2)

B) Yes, because if x ≤ y, then y ≥ x

C) Yes, because if x ≤ y and y ≤ x, then x = y

D) No, because it is not reflexive

39. Consider the relation R on the set of real numbers defined by x R y if x ≤ y. Is this relation reflexive?

A) Yes, because x ≤ x for all real numbers x

B) No, because x ≤ y does not imply y ≤ x

C) Yes, because if x ≤ y and y ≤ z, then x ≤ z

D) No, because not all real numbers are less than or equal to themselves

40. What is an equivalence relation?

A) A relation that is reflexive, symmetric, and transitive

B) A relation that is reflexive and symmetric

C) A relation that is symmetric and transitive

D) A relation that is reflexive and transitive

41. If a relation R on set A is transitive, what condition must hold for any elements a, b, c ∈ A?

A) If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R

B) If (a, b) ∈ R and (a, c) ∈ R, then (b, c) ∈ R

C) If (a, b) ∈ R and (c, b) ∈ R, then (a, c) ∈ R

D) If (a, a) ∈ R and (b, b) ∈ R, then (c, c) ∈ R

42. A relation R on a set A is called symmetric if for every a, b ∈ A, whenever (a, b) ∈ R, then which of the following must also be true?

A) (b, a) ∈ R

B) (a, a) ∈ R

C) (b, b) ∈ R

D) (a, b)⁻¹ ∈ R

43. Which type of relation is defined as a relation R on a set A such that for every element a ∈ A, (a, a) ∈ R?

A) Reflexive relation

B) Symmetric relation

C) Transitive relation

D) Irreflexive relation

44. If R is a relation on set A, what does the notation 'a R b' signify?

A) The ordered pair (a, b) is in the relation R

B) The ordered pair (b, a) is in the relation R

C) a is an element of b

D) b is an element of a

45. What is a binary relation on a set A?

A) A relation from set A to set A

B) A relation from set A to any other set

C) A relation from any set to set A

D) A subset of A

46. If A = {1, 2} and B = {3, 4}, how many possible relations are there from A to B?

A) 4

B) 8

C) 16

D) 256

47. What is the definition of a relation R from a set A to a set B?

A) A subset of the Cartesian product A x B

B) A subset of the Cartesian product B x A

C) The Cartesian product A x B itself

D) A subset of A union B

Relations and types of relations - Question Bank