I am currently in abstract mathematics. I am unclear on how to make a formal equivalence relation proof. I know I must prove reflexive, transitive, and symmetric, but I am not sure the formal set up or even how to for my specific example.
I have to define the relation $\sim$ on $Z \times Z-\{0\}$ by $(a,b) \sim (c,d) \Leftrightarrow ad=bc$ and we must prove that it's an equivalence relation.
Please let me know if you can help!
~ is reflexive $\iff$ $(a,b)$ ~ $(a,b)$ for all $(a,b)$ $\iff$ $ab = ba$ for all $a$ and all $b \ne 0$.
Can you prove that? (Hint: It's trivial)
~ is symmetric $\iff $ $(a,b)$ ~ $(c,d)$ means $(c,d)$~$(a,b)$ $\iff$ $ad = bc$ means $cb = da$. Can you prove that? (Hint: it's trivial)
~ is transitive $\iff$ $(a,b)$~$(c,d)$ and $(c,d)~(e,f)$ means $(a,b) $~$(e,f)$ $\iff$ $ad = bc$ and $cf = de$ means $af=be$ if $b,d,f \ne 0$. Can you prove that? (Hint: It's not trivial but it is not very hard.)