Answer to Question #288610 in Real Analysis for Faisal

Question

If y is not zero and x, y belongs to Q then prove that x/y belongs to Q

Answer 1

Since $y\in\mathbb{Q},y≠0$ and $x\in\mathbb{Q}$

Note that $y≠0,y\in\mathbb{Q}$ $\implies\frac{1}{y}\in\mathbb{Q}$

Since $x\in\mathbb{Q}$ and $y\in\mathbb{Q}$

$\implies xy\in\mathbb{Q}$

Then $\frac{1}{y}.(xy)=(\frac{1}{y}.x)y=(\frac{x}{y})y=x\in\mathbb{Q}$

Since set of rational numbers is closed under multiplication

$\implies\frac{x}{y}\in\mathbb{Q}$