I am trying to write the Reflective Property of Inverse Functions in mathematical notation based on this description:
The graph of $f$ contains the point $(a,b)$ if and only if the graph of $f^{-1}$ contains the point $(b,a)$.
Because the definition mentions "the graph of (a function)", I wish to use some notation to express the graph of a function. On Wikipedia's Graph of a function, the notation used is $G(f)$.
I would then write:
$(a,b)\in G(f) \iff (b, a) \in G(f^{-1})$
However, as a programmer, I would rather use "dot notation", since I think of a function as a mathematical "object" with "members" (object-oriented programming concepts). These members, in my mind, are the domain $f.X$, the codomain $f.Y$, and the graph $f.G$. (The definition of a function as domain X, codomain Y, and graph G can be found in Wikipedia's Function.)
With dot notation, I would write:
$(a,b)\in f.G \iff (b, a) \in f^{-1}.G$
Is there any precedent/examples for dot notation to "access" a "member" of an "object" in math? Is the idea of a mathematical object common? (I see Wikipedia's list of mathematical objects, but it is a very short article.)
Please don't make up new notation for something that's well established. You'll only confuse people. "Dot notation" is not generally used in mathematics, and in particular $f . X$ and $f . Y$ are going to be extremely confusing: while one sometimes has functions whose domain and codomain are $X$ and $Y$, those letters are not in any way reserved for the general notion of domain and codomain.