# Equivilence relations, Reflexivity

• Oct 22nd 2012, 04:17 AM
Srengam
Equivilence relations, Reflexivity
Hi all,

I am given two relations and asked to prove which one is an equivilence relation for integers.

They are;

\$\displaystyle x\$~\$\displaystyle _{1}y\$ if \$\displaystyle x^{2}+y^{2}\$ is divisible by 5.

\$\displaystyle x\$~\$\displaystyle _{2}y\$ if \$\displaystyle x^{2}-y^{2}\$ is divisible by 4.

The problem i'm having is finding which one has reflexivity.

for example, \$\displaystyle 1^{2}+1^{2}=2\$ which is not divisble by 5.

and, \$\displaystyle k^{2}-k^{2}=0\$ for all \$\displaystyle k\$

So, is zero divisible by 4?
• Oct 22nd 2012, 04:29 AM
Srengam
Re: Equivilence relations, Reflexivity
Think I've got,

zero divided by any integer is zero, but since zero is also an integer then it holds.

Am I right?
• Oct 22nd 2012, 04:42 AM
emakarov
Re: Equivilence relations, Reflexivity
You are right. Zero is divisible by any nonzero integer.
• Oct 22nd 2012, 04:44 AM
Plato
Re: Equivilence relations, Reflexivity
Quote:

Originally Posted by Srengam
Think I've got,
zero divided by any nonzero integer is zero, but since zero is also an integer then it holds.
Am I right?

That is correct.