# Usage of “Any” in Math

One must be careful when using the word “any” in mathematical writing because in certain contexts, “any” is ambiguous and can mean both “every” and “at least one”. Consider the following question:

Does $x^2 = 0$ hold for

any$x \geq 0$?

There are two correct answers:

- We can rewrite the question as, “does there exist
*at least one*$x \geq 0$ such $x^2 = 0$?” The answer is “yes” because $x = 0$ satisfies $x^2 = 0$. - Alternatively, we can rewrite the question as, “does
*every*$x \geq 0$ satisfy $x^2 = 0$?” The answer is “no.” If $x = 1$, then $1^2 \neq 0$.

Thus, do not use “any” when what you mean is “all.” Use “for all” or “for each” instead.