axiomatic sets 13 (functions)

We can get a definition of a function,
adding conditons to the definition of a relation.

As a relation is a set, a function is a subset of a relation.

'$F$' is a function if and only if '$F$' is a relation and
\[ \forall x\forall y\forall z[xFy\wedge xFz\rightarrow y=z] \]

It means there exists only one $y$ in a function $xFy$  for giving $x$.

This definition of a function is equivalent to
\[ \forall z[z\in F \rightarrow E!(y)[z=<x,y>\rightarrow xFy]]  \]
,where $<x,y>$  is an ordered pair.

In general, the realtion $\lt(\gt)$ are not a function
because when $x\lt(\gt) y$ and $x\lt(\gt) z$ , we are not able to say $y=z$ .

You must know that in standard notations we write $xFy$  $y=F(x)$ ,
namely $y$  is $F$ of $x$ .

We are able to define a funtion as
\[ F(x)=y\leftrightarrow [E!z[xFz]\wedge xFy]\vee [\neg(E!z[xFz])\wedge y=0] .   \]

In this case, for example, if
\[ F=\left\{<x,y> ; <1,2>,<1,3>,<3,4>,<4,4>    \right\}  \]
,then
\[ F(1)=0,\qquad F(2)=0,\qquad F(3)=4,\qquad F(4)=4 .  \]

You know $\left\{ \right\}=0 $ .