Give a good reason to define a function from A to B as a triple (F, A, B) rather than a functional set of pairs with domain A and image included in B.

Question

The operative part of this question is "good reason": either an example or an argument, without preconceptions or fallacies. The object is comparing two definitions for "a function $f$ from $A$ to $B$", usually introduced by writing $f\colon A \to B$.

Definition I: a functional set of pairs with domain $A$ and image included in $B$.

Definition II: a triple $(F, A, B)$ where $F$ is a functional set of of pairs with domain $A$ and image included in $B$.

Answer 1

The good reason is that this gives us the Category Set of sets (as objects and functions as morphisms).

Answer 2

Consider a disk $D$ in the plane and its boundary circle $C$. The identity function $I$ of $C$ can be regarded as a function $I:C\to C$ and also as a function $I:C\to D$. Under your suggested set-up, these would be the same function, because they are the same set of ordered pairs. Under the alternative "triple" set-up, they are different functions because they have different codomains. In the context of topology, it is much more convenient to regard them as different, because they induce different maps of homology, cohomology, and homotopy groups --- different even in the sense of your suggested set-up.

Answer 3

Perhaps, for a start, we should comment on the idea of "definition" here.

The set-theoretic orthodoxy, at least in its baldest form, identifies a one-place function $f\colon X \to Y$ with its graph $\Gamma_{\!f}$ (its extension, the set of ordered pairs $\langle x, y\rangle$ such that, as we'd ordinarily say, $f(x) = y$ for $x \in X, y \in Y$). And likewise it counts any functional set $\Gamma$ of pairs $\langle x, y\rangle$ where $x \in X$, $y \in Y$ as being an one-place function $f_{\Gamma}$ with arguments among $X$ and values among $Y$. However, there are familiar compelling reasons against such an outright identification claim.

1. In fact even the most enthusiastic set-theorist doesn't really endorse the identification across the board. Take for example the function defined over sets, which maps a set to its singleton. The pairs $\langle x, \{x\}\rangle$ are too many to form a set. So that function lacks a graph; hence not every function is identical to its graph.
2. Setting that damaging point aside and concentrating on sufficiently `small' functions, the association of a function with its graph involves an arbitrary choice of representational scheme (and such arbitrariness precludes us from making an outright identification claim). For a non-artificial example, some textbooks model a two-place function by the set of tuples $\langle y, x\rangle$ where $f(x) = y$, rather than by the set of tuples $\langle x, y\rangle$, and evidently this choice works just as well.
3. The orthodoxy treats an one-place function $f$ with arguments and values both in $X$ as just a special case of an binary relation $R$ over $X$. But identifying a function with a relation in this way is a type-confusion. A function doesn't hold of some object or objects as a relation does: rather it maps some object or objects as arguments to some object as value. And holding of and mapping to are quite different logical features. (I recall Tim Gowers stressing this point somewhere).
4. Continuing the typing theme, a one-place function is as Frege puts it "unsaturated", has -- so to speak -- an empty slot waiting to take an argument. A set of pairs doesn't. (To "apply" a function-as-graph $\Gamma$ to an object, we'll need a binary evaluation function which takes a graph $\Gamma$ and an object $x$, and returns the value $y$ if $\langle x, y\rangle \in \Gamma$). Sets are the wrong logical type of thing to actually be functions.

All these are very familiar points. I'm just banging on about them to remind us that when we talk about "defining" a function as a set, we aren't defining-in-the-sense-of-revealing-the-true-nature-of. Rather we have to be thought of as providing a model or realisation or implementation, chosen as convenient for current purposes.

Which brings us (at last!!) to the question as posed. Neither "definition" of a function should be thought of as being in the business of identifying the true nature of functions. Both are useful models or realisations or implementations for various purposes; which it is appropriate to use depends on the purposes.

Set theorists have got on perfectly well for their purposes with the first definition. Andreas Blass gives a lovely example in his answer of why topologists will want to do things differently. More generally, category theorists want every arrow to have a determinate source and target. And so when dealing with categories of structured sets where the arrows are functions, want to think of these functions as having determinate targets/co-domains. The pure graph model of a function doesn't fix the co-domain. Hence for them, the triple model is a better one to go for. Different strokes for different folks. But that's the nature of the modelling game.