Vaidikalaya

MCQ On Relations

Q1. Which of these is not a type of relation?.
  1. Reflexive
  2. Surjective
  3. Symmetric
  4. Transitive

Answer: b, Surjective

Solution: Surjective is not a type of relation. It is a type of function. Reflexive, Symmetric and Transitive are types of relations.

Q2. An Equivalence relation is always symmetric.
  1. true
  2. false

Answer: a, true

Solution: The given statement is true. A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive. Hence, an equivalence relation is always symmetric.

Q3. Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}..
  1. R = {(1, 2), (1, 3), (1, 4)}
  2. R = {(1, 2), (2, 1)}
  3. R = {(1, 1), (2, 2), (3, 3)}
  4. R = {(1, 1), (1, 2), (2, 3)}

Answer: b, R = {(1, 2), (2, 1)}

Solution: This relation is symmetric because for every (a,b), (b,a) exists. It is not reflexive or transitive.

Q4. Which of the following relations is transitive but not reflexive for the set S={3, 4, 6}?.
  1. R = {(3, 4), (4, 6), (3, 6)}
  2. R = {(1, 2), (1, 3), (1, 4)}
  3. R = {(3, 3), (4, 4), (6, 6)}
  4. R = {(3, 4), (4, 3)}

Answer: a, R = {(3, 4), (4, 6), (3, 6)}

Solution: This relation is transitive since (3,4) and (4,6) imply (3,6), but not reflexive as (3,3),(4,4),(6,6) are missing.

Q5. Let R be a relation in the set N given by R={(a,b): a+b=5, b>1}. Which of the following will satisfy the given relation?.
  1. (2,3) ∈ R
  2. (4,2) ∈ R
  3. (2,1) ∈ R
  4. (5,0) ∈ R

Answer: a, (2,3) ∈ R

Solution: For (2,3), 2+3=5 and b=3>1, so it satisfies the relation.

Q6. Which of the following relations is reflexive but not transitive for the set T = {7, 8, 9}?.
  1. R = {(7, 7), (8, 8), (9, 9)}
  2. R = {(7, 8), (8, 7), (8, 9)}
  3. R = {0}
  4. R = {(7, 8), (8, 8), (8, 9)}

Answer: a, R = {(7, 7), (8, 8), (9, 9)}

Solution: Every element is related to itself, so it is reflexive, but transitivity is not defined as there are no other pairs.

Q7. Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I1, I2): I1 is parallel to I2}. What is the type of given relation?.
  1. Reflexive relation
  2. Transitive relation
  3. Symmetric relation
  4. Equivalence relation

Answer: d, Equivalence relation

Solution: The relation "is parallel to" is reflexive, symmetric, and transitive — hence an equivalence relation.

Q8. Which of the following relations is symmetric and transitive but not reflexive for the set I = {4, 5}?.
  1. R = {(4, 4), (5, 4), (5, 5)}
  2. R = {(4, 4), (5, 5)}
  3. R = {(4, 5), (5, 4)}
  4. R = {(4, 5), (5, 4), (4, 4)}

Answer: d, R = {(4, 5), (5, 4), (4, 4)}

Solution: It is symmetric since (4,5) and (5,4) exist, and transitive as (4,5),(5,4), (4,4). Not reflexive as (5,5) missing.

Q9. (a,a) ? R, for every a ? A. This condition is for which of the following relations?.
  1. Reflexive relation
  2. Symmetric relation
  3. Equivalence relation
  4. Transitive relation

Answer: a, Reflexive relation

Solution: A relation is reflexive if every element is related to itself.

Q10. (a1, a2) ? R implies that (a2, a1) ? R, for all a1, a2 ? A. This condition is for which of the following relations?.
  1. Equivalence relation
  2. Reflexive relation
  3. Symmetric relation
  4. Universal relation

Answer: c, Symmetric relation

Solution: A relation is symmetric if (a,b) ? R implies (b,a) ? R.

Q11. The binary relation {(1,1), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2)} on the set {1, 2, 3} is __________.
  1. reflexive, symmetric and transitive
  2. irreflexive, symmetric and transitive
  3. neither reflexive, nor irreflexive and not transitive
  4. irreflexive and antisymmetric

Answer: c, neither reflexive, nor irreflexive and not transitive

Solution: The relation is not reflexive (missing (3,3)), not symmetric ((2,1) but no (1,2)), and not transitive ((2,3) and (3,1) but no (2,1)). Hence neither reflexive nor transitive.

Q12. Determine the partitions of the set {3, 4, 5, 6, 7} from the following subsets..
  1. {3,5}, {3,6,7}, {4,5,6}
  2. {3}, {4,6}, {5}, {7}
  3. {3,4,6}, {7}
  4. {5,6}, {5,7}

Answer: b, {3}, {4,6}, {5}, {7}

Solution: Partition means disjoint, non-empty subsets whose union equals the set. Only {3}, {4,6}, {5}, {7} satisfies these conditions.

Q13. Determine the number of equivalence classes that can be described by the set {2, 4, 5}..
  1. 125
  2. 5
  3. 16
  4. 72

Answer: b, 5

Solution: Each equivalence relation corresponds to a partition. The number of partitions of a 3-element set is 5 (Bell number B? = 5).

Q14. How many binary relations are there on a set S with 9 distinct elements?.
  1. 290
  2. 2100
  3. 281
  4. 261

Answer: c, 281

Solution: For n elements, number of binary relations = 2n*n For n = 9, it is 29*9 .

Q15. _________ number of reflexive relations are there on a set of 11 distinct elements..
  1. 2110
  2. 2121
  3. 290
  4. 2132

Answer: a, 2110

Solution: Let A be a set consists of n distinct elements. There are 2(n*n)-n number of reflexive relations that can be formed. So, here the answer is 2(11*11)-11 = 2110.

Q16. The number of reflexive as well as symmetric relations on a set with 14 distinct elements is __________.
  1. 4120
  2. 270
  3. 3201
  4. 291

Answer: d, 291

Solution: Let A be a set consists of n distinct elements. There are 2(n*(n-1))/2 number of reflexive and symmetric relations that can be formed. So, here the answer is 214*(14-1)/2 = 291.

Q17. The number of symmetric relations on a set with 15 distinct elements is ______.
  1. 2196
  2. 250
  3. 2320
  4. 278

Answer: a, 2196

Solution: Let S be a set consists of n distinct elements. There are 2(n-1)*(n-1) number of reflexive and symmetric relations that can be formed. So, here the answer is 2(15-1)*(15-1) = 2196.