Question

Check whether the relation R ={(x, y)∈N×N | xy is the square of an integer} is an equivalence relation on N.

Answer 1

Let us show that the relation $R =\{(x, y)\in\N\times \N\ |\ xy \text{ is the square of an integer}\}$ is an equivalence relation on $\N$.

Sinse for any $a\in\N$ we have that $a\cdot a=a^2$, we conclude that $(a,a)\in R$, and hence the relation is reflexive.

If $(x,y)\in R,$ then $xy=n^2$ for some $n\in\N.$ It follows that $yx=n^2,$ and thus $(y,x)\in R.$ Therefore, the relation $R$ is symmetric.

If $(x,y)\in R$ and $(y,z)\in R$, then $xy=n^2$ and $yz=m^2$ for some $n,m\in\N.$ It follows that $(xy)(yz)=n^2m^2,$ and hence $xz=(\frac{nm}{y})^2\in\N.$ We conclude that $(x,z)\in R,$ and the relation $R$ is transitive.

We conclude that the relation $R$ is an equivalence relation on $\N$.