Question

Let $R$ be a relation defined on $N$ as $R=\left\{\right(x,y):y=2x+3,x,y\in N\}$ then which of the following is true

• A. Relation $R$ is reflexive
• B. Relation $R$ is symmetric
• C. Relation $R$ is one-one function
• D. Relation $R$ is not a function

Answer 1

The correct option is C Relation $R$ is one-one function

$y=2x+3$
Let $a\in N$ and $(a,a)\in R$
$\Rightarrow 2a+3=a\Rightarrow a=-3\notin N$
So, $R$ is not reflexive.

Let $(a,b)\in R\Rightarrow b=2a+3\Rightarrow a=\frac{b-3}{2}$
Clearly, $(b,a)\notin R$
So, $R$ is not symmetric.

Here, for each $x\in N$ there is unique $y\in N.$
Hence, $R$ is a function.

Let $f\left(x\right)=2x+3$
and $f\left(x_1\right)=f\left(x_2\right)$
$\Rightarrow 2x_1+3=2x_2+3$
$\Rightarrow x_1=x_2$
Hence, $R$ is one-one function.