Is this correct approach to proving $xy=0$?

Question

Is this a sufficient way of proving that for for all real numbers $x$ and $y$, if $xy=0$ then $x=0$ or $y=0$?

My attempt:

$$xy=0$$ $$y=\frac{0}{x}$$

$$x=\frac{0}{y}$$

Is this a correct approach using axioms of real numbers?

Answer 1

Let $x$ and $y$ be two real numbers and suppose $xy=0$. There are two cases:

• $x=0$;
• $x\ne 0$.

If $x=0$, then "$x=0$ or $y=0$" is true.

If $x\neq 0$, then $x^{-1}$ exists and multiplying this number on both sides of $xy=0$ gives $y=0$, which implies that "$x=0$ or $y=0$" is true.