Show that R is an equivalence relations?

Question

If we define R in Z × (N − {0}) such that (r, s)R(w, z) if and only if ws = zr.

how do I demonstrate R is a equivalence relation?

Answer 1

Specifically, our equivalence relation $R$ was defined by: $\frac{w}{z} R \frac{r}{s}$ if and only if $ws=zr$.

Let’s show that $R$ is an equivalence relation.
Proof:
$(1)$ Reflexive: For any $x$ and $y$, $xy = xy$. So the definition of $R$ implies that $\frac{x}{y} R \frac{x}{y}$.
$(2)$ Symmetric:if $\frac{w}{z} R \frac{r}{s}$ then $ws = zr$ ,so $ zr = ws$ ,so $ rz = sw $ which implies that $\frac{r}{s} R \frac{w}{z}$.
$(3)$ Transitive: Suppose that $\frac{w}{z} R \frac{r}{s}$ and $\frac{r}{s} R \frac{x}{y}$. By the definition of R, we have, $ws=rz$ and $ry=xs$. We then have $wsy=rzy$ and $ryz= xsz$. Since, $rzy=ryz$, we have, $wsy=xsz$. By simple manipulation, we have, $wy=xz$ which gives $\frac{w}{z} = \frac{x}{y}$ satisfying the relation $R$.

Since $R$ is reflexive, symmetric, and transitive, it is an equivalence relation.