Question

Let $A=\left\{2,3,5,6\right\}$. Then which of the following relations is transitive only?

• A. $R=\left\{\left(2,3\right),\left(6,6\right)\right\}$
• B. $R=\left\{\left(2,2\right),\left(3,3\right),\left(5,5\right),\left(6,6\right),\left(2,3\right),\left(3,5\right),\left(2,6\right)\right\}$
• C. $R=\left\{\left(2,6\right),\left(6,2\right)\right\}$
• D. $R=\left\{\left(2,3\right),\left(3,5\right),\left(2,5\right)\right\}$

Answer 1

The correct option is D

$R=\left\{\left(2,3\right),\left(3,5\right),\left(2,5\right)\right\}$

Explanation for correct answer:

Option $\left(D\right)$: $R=\left\{\left(2,3\right),\left(3,5\right),\left(2,5\right)\right\}$

A relation $R$ on set $A$is said to be transitive if $\left(a,b\right)\in R,\left(b,c\right)\in R,\left(a,c\right)\in R$

The given set $A=\left\{2,3,5\right\}$ where if $a=2,b=3,c=5$

we get, $R=\left\{\left(2,3\right),\left(3,5\right),\left(2,5\right)\right\}$ such that $\left(a,b\right)\in R\left(b,c\right)\in R\left(a,c\right)\in R$

Thus, the relation $R=\left\{\left(2,3\right),\left(3,5\right),\left(2,5\right)\right\}$ is transitive.

Therefore, option $\left(D\right)$ is the correct answer.