Is "element of" an a Relation with two Parameters?

Is "element of" a relation with two Parameters in propositional logic like for example father(X,Y) ?

• "element of" is usually used with $\in$ e.g. $1\in\Bbb{N}$ so it says "$1$ is an element of the set of positive integers" (which is true). $\operatorname{father}(x,y)$ is a predicate, which DOES NOT have a truth value. It says "$x$ is a father of $y$" which depends on $x$ and $y$. Commented Feb 5, 2021 at 20:27
• Do you have any reason to doubt it is? The only intuitive one I see is that the element and the set "are not of the same level" contrary, for example, to equality between 2 numbers, 2 sets or 2 functions. But the definition of relation does not depend on this.
– MasB
Commented Feb 5, 2021 at 20:28
• @manooooh I don't understand what you're saying here. The right analogue of "father(x,y)" isn't "$1\in\mathbb{N}$" but, well, "$x\in y$," which is exactly as undetermined. Commented Feb 5, 2021 at 20:39
• @NoahSchweber I didn't mean they are analogue but opposite. I don't think "father(x,y)" is equivalent to $x\in y$. One would usually say "father(Peter, Ana)", meaning that "Peter is the father of Ana" or "Ana is the daughter of Peter". Ana is not a set but a constant, so $y$ is not a set (as you mention) but a variable. Commented Feb 5, 2021 at 22:54

Yes. A relation on $$n$$ sets is ANY subset of the Cartesian product (set of all $$n$$-tuples) of those sets. Frequently for a relation on 2 sets the sets will be the same ($$<$$, $$\subseteq$$, etc), but for $$\in$$ the first set is whatever domain the elements of the second set come from (such as the integers) and the second set is the powerset (set of sets) of that domain.

Symbolically, $$"\in"\subseteq A\times\mathcal{P}(A)$$ where $$A$$ is the domain and $$\mathcal{P}(A)$$ the powerset.