As the title suggests, I am looking at partial functions over a finite set $\mathcal{A}$. Lets denote by $Y_f$ the set of fixpoints of $f$, $C_f$ the set of changed values of $f$, and $B_f$ the points where $f$ is undefined.
$Y_f=\{x \in \mathcal{A} | f(x)=x\} $
$C_f=\{ x \in \mathcal{A} | f(x) \neq x \} $
$B_f=\{ x \in \mathcal{A} | f(x)=\bot \} $
We know this for the cardinalities of the sets involved;
$\mid\mathcal{A}\mid = \mid Y_f\mid + \mid C_f\mid + \mid B_f\mid$
I want to find relationships between these sets and the sets created from a composition of two functions, $f$, $g$. The Bottom set is easy. Note that we assume $f(\bot)=\bot$.
$B_{f\circ g} = B_g \cup g^{-1}(B_f)$
where $g^{-1}(...)$ is the inverse image.
I am stuck trying to find similar relationships for the other two sets. Finding for one of two and using the above to characterize the third in terms of $\mathcal{A}$ is also acceptable. The goal concerns the cardinalities of these sets, but the only way I have found so far to tackle this problem is by looking at the actual sets. What I want to find is some criterion as to when $\mid C_{f \circ g} \mid \leq max\{\mid C_f \mid, \mid C_g \mid \} $, or something along those lines.