Question

If in $N\times N$, $R$ is a relation defined by the formula $(x,y)R(p,q)$ if and only if $x+q=y+p$, show that R is an equivalence relation.

Answer 1

A relation R is said to be in equivalence realtion if it is reflexive,symmetric and transitive.

Reflexive:

If $(x,y)\in N\times N$

$x+y=y+x$

$(x,y)R(x,y)$

$(x,y)\in N\times N$

R is reflexive.

Symmetric:

If $(x,y)R(p,q)$ then,

$x+q=p+y$

$p+y=x+q$

$(p,q)R(x,y)$

R is symmetric.

Transitive:

If $(x,y)R(p,q)$ and $(p,q)R(r,s)$ then,

$x+q=y+p$

$p+s=q+r$

$x+s=y+r$

$(x,y)R(r,s)\in N\times N$

R is transitive

So, R is in equivalence relation