Solution to Check whether the relation R ={(x, y)∈N×N | xy is the square of an integer} … - Sikademy
Author Image

Archangel Macsika

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

The Answer to the Question
is below this banner.

Can't find a solution anywhere?


Get the Answers Now!

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question

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.

Related Answers

Was this answer helpful?

Join our Community to stay in the know

Get updates for similar and other helpful Answers

Question ID: mtid-5-stid-8-sqid-2927-qpid-1626