Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
I am trying to prove the statement in the form:
$\exists x \in S, \forall y \in \emptyset, P(x)$
where $P(x): x > y$ and $S$ is an arbitrary non-empty set.
My proof looks something like:
Let $x =$ some number in set S. Let $y \in \emptyset$. My intuition is that since we cannot check whether $x > y$ since $y$ is an empty set, the statement is vacuously true. My questions is, how do I write this "intuition" in a rigorous way?
You could restate the claim as $\exists x\in S\neg\exists y\in\emptyset\neg P(x)$, which is trivial for each $x\in S$.
This reduces to: $\exists x \colon y \in \{\} \implies P(x)$.
But $y \notin \{\}$.
The truth table of material implication/conditionals ('IF') definitionally maps False to True.
QED.
