Mathematics Questions and Answers – Types of Relations

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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

Answer: b
Explanation: Surjective is not a type of relation. It is a type of function. Reflexive, Symmetric and Transitive are type of relations.
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2. An Equivalence relation is always symmetric.
a) True
b) False
View Answer

Answer: a
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

Answer: b
Explanation: A relation in a set A is said to be symmetric if (a1, a2)∈R implies that (a1, a2)∈R,for every a1, a2∈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 (a1, a2)∈R and (a2, a3)∈R implies that (a1, a3)∈ R for every a1, a2, a3∈R.
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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

Answer: a
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

Answer: a
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
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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

Answer: a
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 (a1, a2)∈R and (a2, a3)∈R implies that (a1, a3) ∈ R for every a1, a2, a3 ∈ R.

7. 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?
a) Reflexive relation
b) Transitive relation
c) Symmetric relation
d) Equivalence relation
View Answer

Answer: d
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 I1 is parallel to I1 i.e. (I1, I2)∈R. Hence, it is a reflexive relation.
Symmetric: Now if a line I1 || I2 then the line I2 || I1. Therefore, (I1, I2)∈R implies that (I2, I1)∈R. Hence, it is a symmetric relation.
Transitive: If two lines (I1, I3) are parallel to a third line (I2) then they will be parallel to each other i.e. if (I1, I2) ∈R and (I2, I3) ∈R implies that (I1, I3) ∈R.
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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

Answer: d
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

Answer: a
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.
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10. (a1, a2) ∈R implies that (a2, a1) ∈ R, for all a1, a2∈A. This condition is for which of the following relations?
a) Equivalence relation
b) Reflexive relation
c) Symmetric relation
d) Universal relation
View Answer

Answer: c
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.

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter