Consider the proposition $f(\alpha)=g(\alpha)$, if one were to write it using logical connectives and the set membership predicate how will it be?

Question

$(((\alpha,\beta)\in f)\land((\alpha,\gamma)\in g))\rightarrow(\beta = \gamma)$ ?

$((\alpha,\beta)\in f)\leftrightarrow((\alpha,\beta)\in g)$ ?

Some other way?

I ask because of this: If $\alpha$ weren't to have an image defined under $f$ (or $g$), if $f(\alpha)$ (or $g(\alpha)$) "didn't exist". will the proposition be true, false or a meaningless statement (only classical logic)?

Answer 1

Assuming that you have that $f$ and $g$ are at least functional:

$(((\alpha,\beta) \in f) \land ((\alpha,\gamma) \in f)) \to \beta = \gamma$ (And same for $g$)

Then if the functions are total:

$\forall \alpha \exists \beta (\alpha,beta) \in f$ (same for $g$)

then the two definitions you provide end up being the same, or at least they can be derived from each other.

But if you don't necessarily have total functions, then I would treat a claim like $f(\alpha) = g(\alpha)$ as saying that they are defined ... I would not consider this claim to be true if both values are undefined. And, given that you don't assume totality, I would then use:

$\exists \beta (((\alpha,\beta) \in f) \land ((\alpha,\beta) \in g))$

in other words: neither of the two, since both would be true if the values would be undefined.