Question

On the set of natural numbers the relation R defined by " xRy if x + 2y = 1" . Discuss the relation for reflexivity, symmetricity, transitivity and equivalence relation​

Answer 1

Let $\displaystyle\sf{x=y}$ then,

$\displaystyle\sf{\longrightarrow x+2x=1}$

$\displaystyle\sf{\longrightarrow 3x=1}$

$\displaystyle\sf{\longrightarrow x=y=\dfrac {1}{3}}\notin\mathbb{N}$

Hence R is not reflexive.

Assume $\displaystyle\sf {(x,\ y)\in R.}$

From definition,

$\displaystyle\sf{\longrightarrow x=1-2y}$

Assume $\displaystyle\sf {(y,\ x)\in R}$ so that,

$\displaystyle\sf{\longrightarrow y+2x=1}$

$\displaystyle\sf{\longrightarrow y+2(1-2y)=1}$

$\displaystyle\sf{\longrightarrow y+2-4y=1}$

$\displaystyle\sf{\longrightarrow y=\dfrac {1}{3}}\notin\mathbb{N}$

Hence R is not symmetric.

Assume $\displaystyle\sf{\longrightarrow (x,\ y),\ (y,\ z)\in R.}$

Then,

$\displaystyle\sf{\longrightarrow x+2y=1}$

$\displaystyle\sf{\longrightarrow y=\dfrac {1-x}{2}\quad\quad\dots(1)}$

And,

$\displaystyle\sf{\longrightarrow y+2z=1}$

From (1),

$\displaystyle\sf{\longrightarrow \dfrac {1-x}{2}+2z=1}$

$\displaystyle\sf{\longrightarrow-x+4z=1}$

This implies R is not transitive.

Hence R is not an equivalence relation.