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?
On
If a statement is true "for all", it implies that it is true for a subset or in this case for any one element.
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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 $m \equiv 1 \pmod 2$.
$\forall m$ is usually read for all m or for each m; for any m is a possible paraphrase in some contexts. $\forall x\big(\varphi(x)\big)$ means that for all possible values of $x$ in the domain of discourse, the statement $\varphi(x)$ about $x$ is true. If you think of $\varphi(x)$ as saying that $x$ has some particular property, then $\forall x\big(\varphi(x)\big)$ says that every element in the domain of discourse has that property.
Your understanding of $\exists$ isn’t quite correct, however; $\exists m\big(\varphi(m)\big)$ means precisely that there is at least one thing in the domain of discourse that has the property in question. It does not exclude the possibility that every element of the domain of discourse has it. For example, if we’re talking about integers, $$\exists n(n\text{ is even }\lor n\text{ is odd}\}$$ is a true statement, even though $$\forall n(n\text{ is even }\lor n\text{ is odd}\}$$ is also true.