GO Classes Set Theory And Algebra Practice Set 1 | Question: 20

Question

Let $R$ be the relation on $M=\{1,2,3\}$ with the following diagraph representation:

Then

• A. $R$ is not reflexive, not symmetric, and not transitive
• B. $R$ is transitive but not reflexive
• C. $R$ is an equivalence relation
• D. $R$ is symmetric but not transitive
• E. $R$ is reflexive but not symmetric

Answer 1

The relation $R$ is define using given digraph over the set $M=\left \{ 1,2,3 \right \}$ is:

$R=\left \{(1,1) (2,2) (1,2) (2,1) (3,1) (3,2) \right \}$

• $R$ is not reflexive because $(3,3)$ is not present in $R$
• $R$ is not symmetric relation because $(3,2)\in R$ but $(2,3)\notin R$
• $R$ is transitive relation as $(3,1) (1,2) \in R\implies(3,2)\in R$. we can check other combinations also.
• $R$ is not an equivalence relation (it should be reflexive, symmetric, and transitive) as it is not reflexive and symmetric.

Option (B) is correct.