Determining equivalence class of relations

Question

Let $R$ be the relation on the set of ordered pairs of positive integers such that $ (a,b)R(c,d)$ if and only if $ad=bc $

Find the equivalence class of $(2,3) $

So what I need to do is find matching pairs for (a,b) s.t. (a,b)R(3,4) right? How to write a general expression for these kinds of questions?

Is it ${{(2k,3k) : k C Z+}}$

Also, I'm wondering what is the answer if they just ask "FInd the equivalence class" (without stating an ordered pair)?

Thank you

Answer 1

If you mean $\{(2k,3k):k\in\Bbb Z^+\}$, then yes, that is ${[(2,3)]}_R$, the equivalence class for $(2,3)$ over the relation $R$.

Mathjax: $\{(2k,3k):k\in\Bbb Z^+\}$

Also, $\forall (a,b)\in{\Bbb Z^+}^2~\big(~{[(a,b)]}_R=\{(c,d)\in{\Bbb Z^+}^2:\frac ab=\frac cd\}~\big)$

Answer 2

This is actually something familiar in disguise. Suppose for a moment that $b \neq 0$ and $d \neq 0$ then $a/b = c/d$. You are just finding different ways to write the same fraction: $\frac{1}{2} \frac{2}{4} \frac{3}{6}$ etc . Some care is required with the $0$ cases.

Answer 3

The equivalence class for (2,3) over the relation R $ = \{(x,y) : (2,3)R(x,y) \ and \ x,y \in \Bbb Z+\}$ $$= \{(x,y) : 2y=3x \ and \ x,y \in \Bbb Z+ \}$$