Proving a property of modulus function by exhaustion: $\forall\ x \in\mathbb{ R} : |xy| = |x||y| $

Question

I would like some clarification of a quick proof of the properties of the modulus function to make sure I'm doing the right thing.

$$\forall\ x \in\mathbb{R} : |xy| = |x||y| $$

If I let $ x,y \in\mathbb{ R} $ and prove by exhaustion do I do this by considering when $x \ge 0, y\ge0$ then $|xy| = |x||y|$ and when $x < 0, y<0$ then $|xy| = |-x||-y| = |x||y|$.

Is this the correct way to go about it?

Answer 1

That works, but then to complete the proof, you also need to consider,

• the cases where $\;x \geq 0, \;y\leq 0,\;$ and by symmetry: $\;x \leq 0,\; y\geq 0$.

Since there are only four cases to consider, proof by exhaustion works just fine.

Answer 2

yes, but don't forget the case where one argument is positive and the other is negative. In proofs by exhaustion you must make sure you really did exhaust all cases.

Just for the sake of pleasing the most pedantic among us, the proof will look nicer if you write it like this: Assume $x\ge 0$ and $y\ge 0$. Then $xy\ge 0$, and thus, by definition, $|x|=x$ and $|y|=y$ and $|xy|=xy$. So, $|xy|=xy=|x||y|$.

Similarly for the other cases.