JEE Main - Relations and types of relations Online Test (MCQ Quiz)

1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

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

A) Equivalence relation B) Universal relation C) Empty relation D) Identity relation

21. 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

22. 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

23. 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

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. 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

26. 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

27. 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

28. 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

29. 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.

30. 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.

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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.

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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.

46. 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

47. 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

Relations and types of relations - Online Test

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