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 $x v=y u .$ Show that $R$ is an equivalence relation.
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 $x v=y u .$ Show that $R$ is an equivalence relation.
Clearly, $(x, y)$ $R (x, y)$, $\forall \,\,(x, y) \in A$, since $x y=y x .$ This shows that $R$ is reflexive. Further, $(x, y) R (u, v)$ $ \Rightarrow x v=y u$ $ \Rightarrow u y=v x$ and hence $(u, v) \,R (x, y) .$ This shows that $R$ is symmetric. Similarly, $(x, y) R (u, v)$ and $(u, v)$ $R$ $(a, b) \Rightarrow x v=y u$ and $u b=v a \Rightarrow $ $x v \frac{a}{u}=y u \frac{a}{u} $ $\Rightarrow x v \frac{b}{v}=$ $y u \frac{a}{u} \Rightarrow $ $x b=y a$ and hence $(x, y) \,R (a, b) .$ Thus, $R$ is transitive. Thus, $R$ is an equivalence relation.
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