For a partial function $p:X\to Y$, we have
- $p^{-1}(A\cap B)=p^{-1}(A)\cap p^{-1}(B)$
- $p^{-1}(A\cup B)=p^{-1}(A)\cup p^{-1}(B)$
- $p^{-1}(A \setminus B)=p^{-1}(A) \setminus p^{-1}(B)$
- $p^{-1}(A \operatorname{\Delta} B)=p^{-1}(A) \operatorname{\Delta} p^{-1}(B)$ where $A \operatorname{\Delta} B := (A \setminus B) \cup (B \setminus A)$
But $p^{-1}(Y)=X$ is only true if $p$ is a total function. Especially $p^{-1}(A^c)=p^{-1}(A)^c$ is only true (even for specific $A$) if $p$ is total, since otherwise $$p^{-1}(Y)=p^{-1}(A\cup A^c)=p^{-1}(A)\cup p^{-1}(A^c) \quad = \quad p^{-1}(A)\cup p^{-1}(A)^c=X$$
We also have
- $A \subseteq B \Rightarrow p^{-1}(A) \subseteq p^{-1}(B)$
- $A' \subseteq B' \Rightarrow p(A') \subseteq p(B')$
- $p(p^{-1}(B)) \subseteq B$
But $A' \subseteq p^{-1}(p(A'))$ is only true (for $A'=X$) if $p$ is a total function.
And then it gets hard to find any further properties absent for partial functions
- $p(A'\cap B') \subseteq p(A')\cap p(B')$ and $p(A'\cap B')=p(A')\cap p(B')$ for injective $p$
- $p(A'\cup B')=p(A')\cup p(B')$
- $p(A' \setminus B') \supseteq p(A') \setminus p(B')$ and $p(A' \setminus B')=p(A') \setminus p(B')$ for injective $p$
- $p(A' \operatorname{\Delta} B') \supseteq p(A') \operatorname{\Delta} p(B')$ and $p(A' \operatorname{\Delta} B')=p(A') \operatorname{\Delta} p(B')$ for injective $p$
- $p(A'^c) \subseteq p(A')^c$ for injective $p$
- $p(A'^c) \supseteq p(A')^c$ for surjective $p$
In a certain sense, the above considerations only identified two properties ($p^{-1}(A^c)=p^{-1}(A)^c$ and $A' \subseteq p^{-1}(p(A'))$) of total functions absent for partial functions, since $p^{-1}(Y)=X$ is just a restatement of the property for being total. Another absent property is suggested at the and of this answer, which is related to operators $\operatorname{op}:\mathcal{P}(Y)\to\mathcal{P}(Y)$ satisfying $A \subseteq B \Rightarrow \operatorname{op}(A) \subseteq \operatorname{op}(B)$. My question is which other properties of this type (i.e. properties which don't explicitly mention elements) are absent for partial functions.