For any integers $m$ and $n$, if $7m+5n=147$, then $m$ is odd or $n$ is odd.
$$Q(m,n) \equiv 7m+5n=147$$
$$∀m∀n: Q(m,n) → \bigl(m \not\equiv 0 \!\!\pmod 2 \lor n \not\equiv 0 \!\! \pmod 2\bigr)$$
Am I right in assuming $\forall$ means "any" in this case? It doesn't seem to make sense to me ($\exists$ to me means "at least one, many, one, all but one, etc; anything less than all but more than none"), but Wikipedia states that $\forall$ can also mean "for any".
Is this correct?
Best Answer
Yes, "for any" means "for all" means $\forall$. "Any" implies you pick an arbitrary integer, so it must be true for all of them.
As for mods: usually, it's not expressed as an operator, but instead as a kind of equivalence relation: $a \equiv b \pmod{n}$ means that $n$ divides $a - b$. So you would write "m is odd" as $m \equiv 1 \pmod 2$.