Question

Let $R$ be relation on the set $N$ be defined by $\left\{(x,y)|x,y\in N,2x+y=41\right\}$. Then $R$ is

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

Answer 1

The correct option is D

None of these

Explanation For The Correct Option:

Determining the nature of $R$,

The given relation $R$ defined on the set $N$ is $\left\{(x,y)|x,y\in N,2x+y=41\right\}$

Checking Reflexivity:

Let $(x,y)=(a,a)\forall a\in N$

$$\begin{gathered} \Rightarrow 2a+a\ne 41 \\ \therefore (a,a)\notin R \end{gathered}$$

Thus, $R$ is not reflexive.

Checking Symmetricity:

Let $(x,y)=(1,39)$

$$\begin{gathered} \Rightarrow 2\times 1+39=41 \\ \therefore (1,39)\in R \end{gathered}$$

Again let $(x,y)=(39,1)$

$$\begin{gathered} \Rightarrow 2\times 39+1=79\ne 41 \\ \therefore (39,1)\notin R \end{gathered}$$

Thus $R$ is not symmetric.

Checking Transitivity:

$$\begin{gathered} \because (a,b)\in R\Rightarrow 2a+b=41 \\ \&(b,c)\in R\Rightarrow 2b+c=41 \end{gathered}$$

But there does not exist such $a$ and$c$ for which

$(a,c)\in R$ so $(a,c)\notin R$

Thus is not transitive.

Hence, option D is the correct answer.