Proof of Reflexivity

suppose m is any integer
now m-m=0
but 3/0
since 0=3(0)
so 3/ (m-m)
hence by definition n of R, mRn

can anyone explain to me the logic behind following these steps to come up with this answer... per se... can anyone explain why the teacher subtracted (m-m) and made it the denominator in the aforementioned problem.

I can't say that this makes any sense to me, care to clarify?

"but 3/0" isn't much of a valid argument as far as I can see?

Are you sure you aren't missing some of what was presented to you? I agree with Capuchin that this doesn't make a lot of sense to me as you have written it.

I believe the teacher is trying to prove that m=m.
Beyond that I don't have much more to give.

m=m anyway regardless, as proven by the reflexive property.

My impression was that this was supposed to be a proof that the reflexive property holds true for integers.

I agree with you. But the reflexive property just seems like a no brainer.

Proof of Reflexivity

suppose m is any integer
now m-m=0
but 3/0
since 0=3(0)
so 3/ (m-m)
hence by definition n of R, mRn

can anyone explain to me the logic behind following these steps to come up with this answer...per se...can anyone explain why the teacher subtracted (m-m) and made it the denominator int he aforementioned problem.
My guess is the following: perhaps the teacher defined R earlier by

mRn if m-n is divisible by 3.


Or "m is related to n by R if the difference m-n is divisible by 3"
"Reflexivity" means that every element is related by R with itself. (It's one of the properties of an equivalence relation). So to prove reflexivity, the teacher needs to prove that for any integer mRm, or in other words that
3 divides (m-m). But (m-m) is just zero, so 3 divides it in a kind of trivial way. So that's done.

I think the tricky bit here is that what the teacher is proving is so simple that it seems strange. But it's helpful to be very clear when you start to prove things, and check of everything precisely.