Is $(a,b) \in \phi \equiv \phi(a,b)$?

Question

Let $A$ and $B$ both be sets such that we have the relation $\phi \subseteq A \times B$. Since a relation is just a predicate of two variables (according to logic of predicates): when expressing that $(a,b) \in \phi$, isn't this equivalent to writing $\phi(a,b)$ (is TRUE)?

I am looking to verify this since I often like to give the defintion of a map to student in the following way (of course, after mentioning that both $A$ and $B$ are sets):

---

A map $\phi : A \to B$ is a relation such that for all $a \in A$ there exists a unique $b \in B$ such that $(a,b) \in \phi$.

---

However, if what I'm asking is true, this could be rewritten as:

A map $\phi : A \to B$ is a relation such that for all $a \in A$ there exists a unique $b \in B$ such that $\phi(a,b)$ (is TRUE).

Answer 1

I'm going to address this question as though we're building mathematics from the ground up. If the question is meant to assume a working knowledge of set theory from the outset, then ignore this answer.

---

Is $(a,b)\in\phi$ logically equivalent to $\phi(a,b)$?

Yes... ish. There are caveats. Technically, the terms "set," "relation" and "function" have distinct meanings that vary depending on context and your choice of foundations. In conversation, "relation," "binary predicate," and "set of pairs" tend to be regarded as the same thing. However, this assumes 1) set theory as a foundation, 2) that every set-theoretic object is a set.

Since "relation" and "function" are not necessarily set-theoretic notions, it is not necessary that $\phi$ be regarded as a set. For instance $\phi$ might be regarded as a function of type $A\times B\to\mathbf{bool}$ (that is a function from $A\times B$ to the type of truth values), a binary relation symbol (as in a first-order theory), or a 2-place predicate (note that predicate and relation are distinct concepts).

The last of these cases is important, as it relates closely to the comments.

...if we examine $A=\{x\mid p(x)\}$... is saying $x\in A$ equivalent to saying $p(x)$ is true, where $p(x)$ is the defining property for $x$ being an element of the set $A$?

In general, for arbitrary predicate $\psi$, the object $\{x\mid \psi(x)\}$ is not a set. This is made evident by Russell's Paradox. It is not difficult to see how the proposed equivalence can lead to a similar situation - take $\phi(a,b)\equiv a=b$, for instance (this leads to $(\phi,\phi)\in \phi$).

To clear things up, set-theorists will sometimes introduce [proper] classes to account for predicates that are not sets. As far as I know, set theories with classes regard every predicate as a class (though, not necessarily a set). In such set theories, the identity $\phi(a,b)\equiv(a,b)\in\phi$ always holds.