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

Step 1: Reflexivity

To prove reflexivity, we need to show that for any element $(x, y)$ in the set A, $(x, y) \mathrm{R}(x, y)$ holds. According to the definition of the relation R, this means we need to check if $x y = y x$.

Step 2: Symmetry

To prove symmetry, we assume that $(x, y) \mathrm{R}(u, v)$ is true, which means $x v = y u$. We then need to show that $(u, v) \mathrm{R}(x, y)$ is also true. This requires demonstrating that $u y = v x$. Since multiplication of integers is commutative, $x v = y u$ can be rewritten as $v x = u y$, which is equivalent to $u y = v x$. Thus, symmetry holds.

Step 3: Transitivity

To prove transitivity, we assume $(x, y) \mathrm{R}(u, v)$ and $(u, v) \mathrm{R}(a, b)$ are true. This means $x v = y u$ and $u b = v a$. Since $x, y, u, v, a, b$ are positive integers, we can divide by them. From $x v = y u$, we get $\frac{x}{y} = \frac{u}{v}$. From $u b = v a$, we get $\frac{u}{v} = \frac{a}{b}$. Combining these, we have $\frac{x}{y} = \frac{a}{b}$, which implies $x b = y a$. This shows that $(x, y) \mathrm{R}(a, b)$, thus transitivity holds.

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