00:01
So we have a point mn that lies in the set z cross c without the origin, okay? and we're gonna define an equivalent relation on mn so that mn are m prime n prime is equal to mn prime m prime n.
00:52
That's times, because they're just integers.
00:56
We know what multiplication of integers means, okay? so two ordered pairs of integers, the second one can't be zero related like this.
01:10
So first of all, is r an equivalence relation, okay? so it's kind of trivially reflexive because if m prime equals m and n prime equals n, then this is clearly gonna be true.
01:40
And it's also trivially symmetric.
01:44
If i interchange the primes and the unprimes, we still get the same thing over here.
01:51
So it's definitely symmetric.
01:58
And the only question is reflexive.
02:02
So if i've got three sets, mn, m prime n prime and m double prime n double prime, okay? so given this, let's see.
02:25
So for it to be reflexive, excuse me, for it to be transitive means that mn are m prime n prime and m prime n prime are m double prime n double prime implies that mn are m double prime n double prime, okay? so the first one of these tells me n m prime equals n prime m.
03:13
And the second one tells me that m prime n double prime equals m double prime n prime.
03:26
We wanna show that we can basically cancel, okay? so if i, so i basically wanna get rid of the n, the primes, right? so if i divide the left -hand sides and the right -hand sides, i get n over n double prime is equal to m double prime over m, that's wrong.
04:07
Let's do it this way.
04:09
N n times m double prime.
04:18
Okay, so m double prime is, we can get from here...
