Exact definition of a function There was a question asked awhile ago concerning the Definition of a function. I have looked at a lot of books and for some reason they pretty much all label a function as $f: A\to B$ with a lower case $f$ where $A$ and $B$ are sets, and then describe $F$ as a subset of $A \times B$.  Why do they do this instead of denoting a function as $F: A\to B$ where $F$ is a subset of $A \times B$? It would make sense to me to write it as a capital letter and be consistent, since functions are sets.
 A: With the language of set theory, we start with the standard definition of a relation $A$ as a set of ordere pairs:

$A \text { is a relation } \leftrightarrow \forall x \ ( x \in A \to  \exists y \ \exists z \ (x= \langle  y,z \rangle))$.

Then we define what is a function:

$f \text { is a function } \leftrightarrow f \text { is a relation and } \forall x \ \forall y \ \forall z \ ((\langle x, y \rangle \in f \land \langle x, z \rangle \in f) \to y=z)$.

At this point, we need a new definition:

$f(x)=y \leftrightarrow [\exists ! z \ (\langle x, z \rangle \in f) \land \langle x, y \rangle \in f ]$.

A: I disagree with the premise that a function is a set. Any function can be expressed as a set, surely, but not every set is a function (actually, most sets are not). Not even the Cartesian product of two sets is a good description of a function, since $A\times B$ will connect several elements of $B$ to the same $A$.
Is a number the same as a sequence?
Similarly, you could claim that a sequence is the same as a number. It's true: Any number can be described by a sequence (infinitely many sequences even), but not every sequence describes a number in this sense. And in the end, with a sequence you can do many things that make no sense with numbers - for example take out individual elements.
The same way, many things that make sense for sets (for example taking their complement) do not make sense for functions when used as sets.
Different ways to look at the same thing
In mathematics, there are often many ways to look at the same thing. A function can be viewed as a set, indeed. It can also be viewed as an infinite-dimensional vector. It can be viewed as an algorithm, a process to produce a result.
The thing is, a function is something rather specific. When you say "let $f$ be a function", you equip $f$ with every part of what defines a function. Saying "let $f$ be a set" or "let $f$ be a vector" would not provide every part, so you would need to go on with "also make sure that $f$ does not combine one element of $B$ to more than one element in $A$", etc.
And as functions play a very important role in mathematics, they got their own name, and a usual notation. Like, for example, an ordinary vector in $\mathbb R^n$, which is of course also a finite sequence, or a function $\{1, 2, ..., n\} \rightarrow \mathbb R$, or, by your definition, a set $\{1, 2, ..., n\} \times\mathbb R$, but when you say "vector in $\mathbb R^n$, everything about it is said.
Similarly, calling $f(x)$ a function is just a way of saying "You know, the kind of set over $A \times B$ where every $B$ has only one partner in $A$, or the kind of vector, that may have infinitely many entries and the index is given by $x$." When you call it a function, everything relevant is said.
Which way is the right one?
Knowing other ways of looking at it (as a set, as a possibly infinite-dimensional vector, as the limit of a series of other functions, or as the result of an algorithm for example) is excellent in certain applications, but I do not suggest taking one particular interpretation as the ultimate one, because for example your set-interpretation would make it very difficult indeed to come up with a notation like $f(x)$, with continuity, derivatives, or adding two functions.
It is often very easy to see what makes up a function from $A$ to $B$, but it is difficult to see when all you have is a bunch of $(a,b)$ tuples.
