Does it make sense to talk about surjectiveness of functions defined as ordered pairs?

Question

I always defined a function as a set of ordered pairs $(x,y)$ such that some property holds, and now that I think about it, I think it doesn't make sense to talk about surjectiveness with this definition.

When defining a function as a tuple $(f,A,B)$ where $f$ is a functional relation, $A=\text{Dom}f$ and $B\supseteq\text{Im}f$ it does make sense to talk about surjectiveness, because $B$ may be a proper superset of $\text{Im}f$, so we need some differentiation to talk about surjective or non-surjective functions with this definition.

But now, as I see it, if we define a function as set of ordered pairs, it doesn't make sense to talk about surjectiveness, but if we define it as a tuple it does make sense, so, the choice of the definition of a function tells wether this concept of surjectiveness makes sense, which should not happen, as a function is a function, and its properties should not change when changing the definition(if the definitions are equivalent, obviously).

So, what am I missing here, or, which is the "correct" definition of a function? Certainly surjectiveness has helped to tackle some problems, but the choice of definition should not change the properties, I think. Am I correct?

Answer 1

With every function $f\colon A\to B$, we can associate its graph $\Gamma_f = \{(x,f(x))\in A\times B\mid x\in A\}$. If you identify function $f$ with its graph $\Gamma_f$, then it doesn't make sense to talk about surjectivity anymore, since you lost information about codomain.

On the other hand if you do remember that $\Gamma_f\subseteq A\times B$, then surjectivity makes sense.

The point is, domain and codomain are essential part of any definition of function. It is not enough just to give a set of ordered pairs.

Answer 2

The abstract definition of a function comes from the definition of a relation. A relation $R$ on two sets $A$ and $B$ is any subset of their Cartesian product, so

$$R\subseteq A\times B$$

If $(x,y)\in R$, then we write $xRy$, read as $x$ is related to $y$.

A relation $f$ is called a function if the following criteria hold:

First, for any $x\in A$, there exists a $y$ such that $xfy$.

Second, if $xfy$ and $xfz$, then $y=z$.

The first criteria says that each element of the domain $A$ must be related to something in the target $B$. The second criteria states that each element in the domain is related to exactly one element in the target. Your definition of a function is intuitively correct, since you describe it as a collection of ordered pairs, which it is by definition of being a relation. The issue is that the function is necessarily a subset of a Cartesian product. This means that we need to define some collection that we call the target set. You could, in theory, be given a function without knowing the target set, and not know if it is surjective or not. For example, if I define the function

$$f(x)=x$$

then this appears to be a surjective function by how it is naturally described. However, if I tell you that the target set is the set of complex numbers, then this is not a surjective function.

Ultimately, a function is generally created to serve some kind of purpose, and so the idea of surjectivity is one we impose to discuss how this purpose is performed. A surjective operation reaches every target we can ostensibly think of (or in any sense care about), while a nonsurjective function does not.

Answer 3

But now, as I see it, if we define a function as set of ordered pairs, it doesn't make sense to talk about surjectiveness, but if we define it as a tuple it does make sense, so, the choice of the definition of a function tells wether this concept of surjectiveness makes sense, which should not happen, as a function is a function, and its properties should not change when changing the definition(if the definitions are equivalent, obviously).

The problem is that the definitions are not equivalent. A function is a function, yes, but a function is just an intuitive notion. There are inequivalent ways of formalizing that notion rigorously. The two definitions are distinct and irreconcilable, and this is something you just need to accept if you want to further your understanding of why we have these definitions in the first place.

So, what am I missing here, or, which is the "correct" definition of a function?

There is no "correct" definition. The two definitions are have their advantages and disadvantages, and depending on the context, you may want to prefer one over the other, but neither is clearly more useful than the other in general. If it helps you feel better, though, some mathematicians like to distinguish between functions, for which they use the first definition you provided, and maps, for which they use the second definition you provided. The distinction is useful in its own right in many contexts, and being able to switch back and forth between the two inequivalent but related notions is flexibility that we appreciate, as it is a virtue.

Certainly surjectiveness has helped to tackle some problems, but the choice of definition should not change the properties, I think. Am I correct?

No. If the definitions are equivalent, then the definition change should not change the object itself. But the definitions are definitely inequivalent, in this case, which is what I think you are failing to see. The second definition specifies a triple $(X,Y,F)$. The first definition defines a function as $F$ itself. So it should be obvious how the definitions are inequivalent.