Let $X = \{(a,b)|a,b \in \mathbb Z, b \neq 0 \}$. Define a relation $\sim$ on $X$ by $(a,b) \sim (c,d)$ iff $ad = bc $.
a) I'm trying to show that ~ is an equivalence relation.
So, is it reflexive?
$(a,b) \sim (a,b) $ iff $ab = ab$, with $b$ not equal to zero
so its reflexive
Is it transitive?
$(a,b) \sim (c,d) $ iff ad = bc and if $(c,d) \sim (e,f)$ then $cf = de$ so, $c = ad/b$ and $d = bc/a$
and we have then $ad/b *f = bc/a *e$
Is it symmetric?
$(a,b) \sim (a,b)$ iff $ab = ab$, same as $ab = ba$, (why?) with $b$ not equal to zero
b) Let $\mathbb Q$ be the set of equivalence classes and denote the equivalence class of $(a,b)$ by $[a,b]$. I'm needing help in showing that the "addition and Multiplication" of $\mathbb Q$ are well defined:
(1) $[a,b] + [c,d] = [ad+bc,bd]$
and
(2)$ [a,b][c,d] = [ac,bd]$
What I have so far:
(1) $[a,b] + ([c,d] +[e,f]) = [a,b] + ([cf,de]) = [ade,bcf]$
but this is true for when ?
(2) If $c =b$ and $d = a$
then, $[a,b][b,a] = [a^2,b^2]$ iff $a^2 = b^2$, which is not well defined, since $a = -5$ and $b$ could be equal to $5$, with $25 = 25$