Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $xv = yu$. Show that $R$ is an equivalence relation.

$1$. Reflexivity: For any $(x, y) \in A$,we have $xy = yx$,which implies $(x, y) R (x, y)$. Thus,$R$ is reflexive.
$2$. Symmetry: If $(x, y) R (u, v)$,then $xv = yu$. This implies $uy = vx$,which is equivalent to $(u, v) R (x, y)$. Thus,$R$ is symmetric.
$3$. Transitivity: Suppose $(x, y) R (u, v)$ and $(u, v) R (a, b)$. Then $xv = yu$ and $ub = va$. From $xv = yu$,we have $\frac{x}{y} = \frac{u}{v}$,and from $ub = va$,we have $\frac{u}{v} = \frac{a}{b}$. Therefore,$\frac{x}{y} = \frac{a}{b}$,which implies $xb = ya$. Thus,$(x, y) R (a, b)$,and $R$ is transitive.
Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.