What's the correct way to define a function? (codomain and surjectivity)

Question

Here's a common definition of a function (for example, Wiki follows this definition):

A relation between sets $A$ and $B$ is any subset $R \subseteq A \times B$. We say that this relation is a function if it satisfies the property $$ (a,b_1) \in R \text{ and }(a,b_2) \in R \implies b_1 = b_2 $$

This definition of a function is good for most purposes. Certainly, we can find the domain and range of a function defined in this way. This also allows us to answer such questions as

Is the set $\{(1,2),(2,5),(3,5)\}$ a function? (answer: yes)

With such a definition, we could certainly go on to define a function's domain and range. The problem with this definition, however, is that it includes no notion of a codomain. This presents a problem when we want to answer a question such as

Are the functions $f:\Bbb R \to \Bbb R$ given by $f(x) = x^2$ and $g:\Bbb R \to [0,\infty)$ given by $g(x) = x^2$ the same function? (answer: no)

On the one hand, the "graphs" of the function are the same. If we are to believe that these functions are merely subsets of a Cartesian product, then we should say that $f = g$ since both are merely the set $\{(x,x^2) : x \in \Bbb R\}$. On the other hand, we would like to say that "the function $g$ is surjective, but the function $f$ is not". If surjectivity is a property of functions, then the fact that $f$ and $g$ do not share this property should mean that $f \neq g$.

So what gives? Is there a setting in which both of these questions are well-posed? If someone has a reference that handles all this well, I would appreciate it.

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Edit: Wiki apparrarently has a discussion of this issue here

Answer 1

Forget about functions for a minute and go back to the definition of a relation.

A relation between a first set $A$ with a second set $B$ is any subset of $A \times B$. The set $A$ is called the domain of the relation and the set $B$ is called the codomain.

By definition, when you look at any relation $\tau$, it is not just a subset of a cartesian product. You have a map,

Domain($\tau$) = $A$
Codomain($\tau$) = $B$

Part of the definition/structure of a relation is this underlying concept of domain and codomain. You can't 'forget it'.

If you have any function $f$ between $A$ and $B$ you can define a new function between $A$ and $f(<A>)$. This is now a surjective function. What do you want to call this function. If there is no danger of getting into trouble, how about $f$?.

So this is part of the definition. If you want you can say,

Consider the 'functional graph' of $y = x^2$ in $\Bbb R \times \Bbb R$.

or

Consider the 'functional graph' of $y = \sqrt x$ in $\Bbb R \times \Bbb R$.

Depending on your audience this might be fine and nobody will get bent out of shape worried about domains and codomains. See wikipedia Codomain; you can 'blame' Nicolas Bourbaki for this.

Answer 2

Upon closer reading, I've found out how the text I'm teaching from resolves this paradox.

Indeed, a function is considered equal to its graph. Rather saying that a function is inherently surjective/onto, the text opts to define the phrase $f$ is "onto $B$" to mean that $f$ is surjective when $B$ is taken as the codomain. If we establish a codomain by writing $f:A \to B$, then "onto/surjective" means "onto $B$".

So, there is indeed a valid framework. We do forego the notion that surjectivity is a property of a function, but I think this is a reasonable compromise.