This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Types of Relations”.

1. Which of these is not a type of relation?

a) Reflexive

b) Surjective

c) Symmetric

d) Transitive

View Answer

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

2. An Equivalence relation is always symmetric.

a) True

b) False

View Answer

Explanation: 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.

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

a) R = {(1, 2), (1, 3), (1, 4)}

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

c) R = {(1, 1), (2, 2), (3, 3)}

d) R = {(1, 1), (1, 2), (2, 3)}

View Answer

Explanation: A relation in a set A is said to be symmetric if (a

_{1}, a

_{2})∈R implies that (a

_{1}, a

_{2})∈R,for every a

_{1}, a

_{2}∈R.

Hence, for the given set A={1, 2, 3}, R={(1, 2), (2, 1)} is symmetric. It is not reflexive since every element is not related to itself and neither transitive as it does not satisfy the condition that for a given relation R in a set A if (a

_{1}, a

_{2})∈R and (a

_{2}, a

_{3})∈R implies that (a

_{1}, a

_{3})∈ R for every a

_{1}, a

_{2}, a

_{3}∈R.

4. Which of the following relations is transitive but not reflexive for the set S={3, 4, 6}?

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

b) R = {(1, 2), (1, 3), (1, 4)}

c) R = {(3, 3), (4, 4), (6, 6)}

d) R = {(3, 4), (4, 3)}

View Answer

Explanation: For the above given set S = {3, 4, 6}, R = {(3, 4), (4, 6), (3, 6)} is transitive as (3,4)∈R and (4,6) ∈R and (3,6) also belongs to R . It is not a reflexive relation as it does not satisfy the condition (a,a)∈R, for every a∈A for a relation R in the set A.

5. 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?

a) (2,3) ∈ R

b) (4,2) ∈ R

c) (2,1) ∈ R

d) (5,0) ∈ R

View Answer

Explanation: (2,3) ∈ R as 2+3 = 5, 3>1, thus satisfying the given condition.

(4,2) doesn’t belong to R as 4+2 ≠ 5.

(2,1) doesn’t belong to R as 2+1 ≠ 5.

(5,0) doesn’tbelong to R as 0⊁1

6. Which of the following relations is reflexive but not transitive for the set T = {7, 8, 9}?

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

b) R = {(7, 8), (8, 7), (8, 9)}

c) R = {0}

d) R = {(7, 8), (8, 8), (8, 9)}

View Answer

Explanation: The relation R= {(7, 7), (8, 8), (9, 9)} is reflexive as every element is related to itself i.e. (a,a) ∈ R, for every a∈A. and it is not transitive as it does not satisfy the condition that for a relation R in a set A if (a

_{1}, a

_{2})∈R and (a

_{2}, a

_{3})∈R implies that (a

_{1}, a

_{3}) ∈ R for every a

_{1}, a

_{2}, a

_{3}∈ R.

7. Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I_{1}, I_{2}):I_{1} is parallel to I_{2}}. What is the type of given relation?

a) Reflexive relation

b) Transitive relation

c) Symmetric relation

d) Equivalence relation

View Answer

Explanation: This is an equivalence relation. A relation R is said to be an equivalence relation when it is reflexive, transitive and symmetric.

Reflexive: We know that a line is always parallel to itself. This implies that I

_{1}is parallel to I

_{1}i.e. (I

_{1}, I

_{2})∈R. Hence, it is a reflexive relation.

Symmetric: Now if a line I

_{1}|| I

_{2}then the line I

_{2}|| I

_{1}. Therefore, (I

_{1}, I

_{2})∈R implies that (I

_{2}, I

_{1})∈R. Hence, it is a symmetric relation.

Transitive: If two lines (I

_{1}, I

_{3}) are parallel to a third line (I

_{2}) then they will be parallel to each other i.e. if (I

_{1}, I

_{2}) ∈R and (I

_{2}, I

_{3}) ∈R implies that (I

_{1}, I

_{3}) ∈R.

8. Which of the following relations is symmetric and transitive but not reflexive for the set I = {4, 5}?

a) R = {(4, 4), (5, 4), (5, 5)}

b) R = {(4, 4), (5, 5)}

c) R = {(4, 5), (5, 4)}

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

View Answer

Explanation: R= {(4, 5), (5, 4), (4, 4)} is symmetric since (4, 5) and (5, 4) are converse of each other thus satisfying the condition for a symmetric relation and it is transitive as (4, 5)∈R and (5, 4)∈R implies that (4, 4) ∈R. It is not reflexive as every element in the set I is not related to itself.

9. (a,a) ∈ R, for every a ∈ A. This condition is for which of the following relations?

a) Reflexive relation

b) Symmetric relation

c) Equivalence relation

d) Transitive relation

View Answer

Explanation: The above is the condition for a reflexive relation. A relation is said to be reflexive if every element in the set is related to itself.

10. (a_{1}, a_{2}) ∈R implies that (a_{2}, a_{1}) ∈ R, for all a_{1}, a_{2}∈A. This condition is for which of the following relations?

a) Equivalence relation

b) Reflexive relation

c) Symmetric relation

d) Universal relation

View Answer

Explanation: The above is a condition for a symmetric relation.

For example, a relation R on set A = {1,2,3,4} is given by R={(a,b):a+b=3, a>0, b>0}

1+2 = 3, 1>0 and 2>0 which implies (1,2) ∈ R.

Similarly, 2+1 = 3, 2>0 and 1>0 which implies (2,1)∈R. Therefore both (1, 2) and (2, 1) are converse of each other and is a part of the relation. Hence, they are symmetric.

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