Contradiction in proof by contraposition and counterexample

Suppose $\frac{x}{y}$ is rational, then $x \in Z$ and $y \in Z$ by definition of rational

False. If $x=y=\frac12$, then $\frac{x}{y}$ is rational, but $x\notin \mathbb Z$ and $y\notin\mathbb Z$.

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Discrete Mathematics

Elementary Number Theory

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