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I'm reading through a few analysis books and I am a little confused by some of the definitions that are given for functions. Some texts define functions to be some subset of the cartesian product of two sets, given that the elements of this subset satisfy the properties of a function. This is intuitively clear to me, but others first define the idea of a relation and then define functions to be a sort of relation that satisfies the typical properties of a function.

This is confusing to me since I have always thought of relations as having truth values and functions as having no truth value. Is it appropriate to think of relations as having truth values and functions as not having truth values? Are the varying definitions compatible or is one wrong? I appreciate any help.

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  • $\begingroup$ A relation $R$ is a set (subset of the cartesian product...): we write $(x,y) \in R$ or $xRy$ meaning that the two "objects" $x$ and $y$ are $R$-related. A function $f$ is a relation that is "functional"; we usually write $f(x)=y$ but this is not incompatible with $(x,y) \in R$. $\endgroup$ Mar 16 '17 at 19:20
  • $\begingroup$ We can write $f(x)=y$ when $(x,y) \in f$ exactly because $f$ is "fucntional", i.e. there is exactly one $y$ such that $(x,y) \in R$. $\endgroup$ Mar 16 '17 at 19:20
  • $\begingroup$ In logic, we have predicate symbols $P(x_1, \ldots, x_n)$ and function symbols $f(x_1, \ldots, x_m)$. The first one is interpreted with a $n$-ary relation on the domain $D$ of the interpretation. $\endgroup$ Mar 16 '17 at 19:22
  • $\begingroup$ A truth value is not given to a relation $R$, but truth values are given to statements of the form $xRy$. Similarly, a truth value is not give to a function $f$, but truth values are given to statements of the form $y=f(x)$. $\endgroup$
    – GEdgar
    Mar 16 '17 at 19:23
  • $\begingroup$ The second one with an $m$-ary function from $D \times \ldots D$ ($m$-times) to $D$. But an $m$-ary function is also an $(m+1)$-ary relation taht is "functional". $\endgroup$ Mar 16 '17 at 19:24
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You are in good company in resisting the identification of a function as a sort of relation. Here's a short excerpt from a useful blog post by the Field's medallist Tim Gowers (I mention his achievement just to point up that this is a post by a top class mathematician, not by e.g. a pernickety philosopher with an amateur interest in these things!):

Just before I move on, let me express particular distaste for any definition that begins, “A function from A to B is a relation such that …” I absolutely hate this. The reason I hate it is that functions and relations are, to any reasonable person, different kinds of things, except that I don’t want to call them things at all, so what I really mean is that they have a different grammar.

To illustrate what I mean by grammar, it’s rules like this.

  1. If $f:A\to B$ and $x$ denotes an element of $A$, then $f(x)$ denotes an element of B.

  2. If P and Q denote statements, then $P\vee Q$ denotes a statement.

  3. If $\sim$ is a relation on $A\times B$, $x$ is an element of $A$, and $y$ is an element of $B$, then $x\sim y$ is a statement.

  4. If $P$ is a property defined on a set $X$ and $x\in X$, then $P(x)$ is a statement.

I’ll talk more about this kind of thing in a later post, but I hope these examples give you the idea. And 1 and 3 demonstrate that the grammar of functions is not the same as the grammar of relations.

And Gowers go on to insist that the fact that we can trade in (much) function talk for relation talk in a way that maps truths to equivalent truths does not mean that functions are "really" relations. And the fact that we can for some purposes usefully trade both function talk and relation talk for talk of their graphs/extensions doesn't mean that functions and relations are "really" sets either (as Gowers also explains).

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In logic we use predicate symbols to express that some relation holds between some number of objects .. and that would indeed be a claim with a truth-value. And we use function symbols to express functions that are used to map objects to other objects ... no claim or truth-value there.

But when we talk about a relation as a set of tuples, and a function as a specific kind of such a relation (one that is right-unique or 'functional'), we really don't make any claims in either case.

Example:

$1<2 $ is a claim involving a relation.

$1+2$ is an operation involving a function ... not a claim

$R_< = \{ (x,y) | x < y \} = \{ (0,1), (0,2), (1,2) , ... \}$ is a relation .. no claim here

$+$ is a function .. and can be expressed as $\{(x,y,z) | x + y = z \} = \{ (0,0,0), (1,2,3), ... \}$ .. again, no claim here

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A function $f$ corresponds to the relation $xRy$ iff $f(x) = y$.

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