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What makes it necessary to define the graph of a function $f:A\rightarrow B$ as $$\{(x,f(x))\mid x\in A\}$$ which makes it a subset of $A\times B$, when this is equal to the function itself, which is defined to be a left-unique and left-defined relation from $A$ to $B$ — and thus a subset of $A\times B$, too?

EDIT: I disagree that with this set-based approach information about the domain and range would get lost, because as I perceived it, there is no definition of a “function”, but rather of a “function from A to B”, in mathematical terms: $f$ is a function from $A$ to $B:\Leftrightarrow f$ is a relation between $A$ and $B \land \forall a\in A\exists! b\in B: (a,b)\in f$.

If we now want to construct some definitions regarding functions, we do not write “Let $f\subset A\times B$”, but rather write “let $f$ be a function from $A$ to $B$”. This makes by definition $f\subset A\times B$, but also preserves the additional information as the domain etc.

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Identifying the graph with the function is a common approach in set theory, especially elementary set theory. It is awkward in other disciplines, where it is more convenient for a function to be a special kind of object which can have properties that are specific to functions (continuity, for example).

A case of this from topology is that if $A$ and $B$ are topological spaces, then the graph of a function can be naturally made into a topological space (it has the subspace topology inherited from the product topology on $A \times B$). So now by making this identification we are saying that a topological space and a function are the same object, which is confusing at best. Similar problems occur in algebra or indeed anywhere where the product has more than just its set-theoretic structure.

Another problem is that the graph does not have the domain conveniently "stored"; it has to be extracted from the projection, whereas the domain of a function is usually regarded as an inherent attribute of the function. Even worse, the codomain isn't accessible from the graph at all; the best way one could define the codomain from the graph would be to identify it with the image. This would be a problem for defining surjectivity and related notions.

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  • $\begingroup$ I do not see any conflicts. I think I somehow get your points, but I need some concrete example in which you use my definition trying to define something like continuity (or even something more trivial) and point out where the formal / logical issue is. Anyway, how would you define a function differently? $\endgroup$ Commented Jan 28, 2015 at 22:36
  • $\begingroup$ Looking at the topology example, I think it should be apparent that the function $f : [0,1] \to [0,1]$, $f(x)=x$ is not really the same as the diagonal $\{ (x,x) : x \in [0,1] \}$. The former is a continuous function; the latter is a closed subset of $\mathbb{R}^2$. At a minimum I would say that a function is the triple $(A,B,f)$ where $A$ is the domain and $B$ is the codomain. There might be more structure depending on the structure of the domain and codomain. $\endgroup$ Commented Jan 28, 2015 at 22:55
  • $\begingroup$ In this triple, what is f? The relation? $\endgroup$ Commented Jan 28, 2015 at 23:08
  • $\begingroup$ @Lukas $f$ is the graph here, yes. Perhaps $f=(A,B,G)$ would be clearer nomenclature. $\endgroup$ Commented Jan 28, 2015 at 23:10
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    $\begingroup$ @Lukas That's not referentially transparent, which is a considerable philosophical problem. $\endgroup$ Commented Jan 28, 2015 at 23:40

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