If I have a given function $f$, is there a standard notation for $f$'s domain, codomain and graph? One that I can use even if $f$ was not explicitly defined?
The definition of a function, as I was taught, is an ordered 3-tuple $\left(A,B,G\right) $ where $A$ is the domain, $B$ is the co-domain, and $G$ is the graph.
For example if I have the function $f=\left(A,B,G\right)=\left(\left\{ 1,2,3\right\} ,\left\{ 4,5,6\right\} ,\left\{ \left(1,4\right),\left(2,5\right),\left(3,4\right)\right\} \right)$
Meaning $\begin{cases} f\left(1\right)=4\\ f\left(2\right)=5\\ f\left(3\right)=4 \end{cases}$
Is there a standard notation similar to this?:
$Domain(f)=\left\{ 1,2,3\right\}$
$Codomain(f)=\left\{ 4, 5, 6\right\}$
$Graph(f)=\left\{ \left(1,4\right),\left(2,5\right),\left(3,4\right)\right\}$
The closest notation I can think of is for $f$'s image: $f[A]$, and using this, $f$'s domain can be (somewhat convolutedly) written as $f^{-1}[f[A]]$. Both of these notations require $A$ to be explicitly named/defined, which is disadvantageous.
As @Xander and @saulspatz mentioned above, it appears the applicable notations are:
$dom(f) = D(f) = Dmn(f)$
and
$ran(f) = R(f) = Rng(f)$