Question

In the set $A=\left\{1,2,3,4,5\right\}$ a relation $R$ is defined by $R=\left\{(x,y)|x,y\in Aandx<y\right\}$. Then $R$ is

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of these

Answer 1

The correct option is C

Transitive

Explanation For The Correct Option:

Determining relation $R=\left\{(x,y)|x,y\in Aandx<y\right\}$ in $A=\left\{1,2,3,4,5\right\}$ is reflexive, symmetric or transitive :

In $A=\left\{1,2,3,4,5\right\}$

Pick $x=2,y=2\in A$ then $2$ is not less than $2$

So, $R$ is not reflexive as ${}_x{R}_y{\nsim }_y{R}_x$ not satisfied

As $\left(x=2,y=3\right)\in R$ but $\left(y=3,x=2\right)\notin R$ then $3$is not less than $2$ so

So, $R$ is not symmetric as ${}_x{R}_y{\ne }_y{R}_x$

Now pick $\left(x=2,y=3\right)\in R\&\left(y=3,z=4\right)\in R$ then $\left(x=2,z=4\right)\in R$

So, $R$ is transitive as ${}_x{R_{Y,}}_Y{R}_Z{\Rightarrow }_x{R}_Z$Satisfied.

Hence, option C is the correct answer.