Let $R$ be the relation on $N\times (N\setminus\{0\})$ defined by $((a, b),(c, d)) \in R$ if $ad = bc$. Prove that $R$ is an equivalence relation.
I'm pretty confused with this problem, mainly because I don't understand the significance of $(N\setminus\{0\})$. Any ideas on how to solve this?
There are 3 things you need to show:
$x\equiv x$ (reflexivity) i.e. $(a,b)\equiv (a,b) \implies ab = ab$ which is true for all $(a,b)$
$x\equiv y \implies y\equiv x$ (symmetry) I am going to leave this and the next one to you.
$x\equiv y,y\equiv z \implies x\equiv z$ (transitivity)
You can just grind through the algebra with little concern what the eqivalence relation actually implies.
However, it might help your intuition if you notice that:
$\frac ab = \frac cd \implies ad = bc$
And then it will make some more sense why $b,d$ cannot equal $0.$
Regardless, it is the same steps to prove the equivalence relation.