Though searching for previous questions returns thousands of results for the query "for any" "for all", none specifically address the following query:
I'm reading a textbook in which one definition requires that some condition holds for any $x,\ x' \in X,$ and right afterwards another definition requires that some other condition holds for all $x,\ x' \in X.$
Is there a difference between 'for any' and 'for all'?
This seems to depend on the context: "For all $x \in X \ P(x)$" is the same as "For any $x∈X \ P(x)$" On the other hand "If for any $x∈X \ P(x)$, then $Q$" means that the existence of at least one $x\in X$ with $P(x)$ implies $Q$, so $P(x)$ doesn't need to hold for all $x \in X$ to imply $Q$.