Description of the image of a function

Let $f:A\to B$ be a map, then the map induced by $f$ is defined as $f^{\to}:\mathrm{P}(A)\to \mathrm{P}(B),\ X\mapsto f(X).$ Here $\mathrm{P}(A)$ denotes the power set of $A.$ We know that $f^{\to}$ satisfys the following propeties:

Property: $\bf 1)$ $f^{\to}(X)=\emptyset\Longleftrightarrow\ X=\emptyset;$

$\bf 2)$ $f^{\to}(X\cup Y)=f^{\to}(X)\cup f^{\to}(Y);$

$\bf 3)$ $f^{\to}(X\cap Y)\subset f^{\to}(X)\cap f^{\to}(Y).$

One may wonder if the properties above could ensure that a function $F:\mathrm{P}(A)\to \mathrm{P}(B)$ is induced by some map $f:A\to B.$ It's very possibly that there could be some counterexamples.

Question: $\bf 1)$ Given a function $F:\mathrm{P}(A)\to \mathrm{P}(B)$ satisfying the properties above, can we always find some map $f:A\to B$ such that $F=f^{\to}?$

$\bf 2)$ If not, can we strengthen the requirements of $F$ to guarantee that $F$ is induced by some map $f:A\to B?$

EDIT: Now I've found proper conditions to ensure that $F$ is induced by a map $f:A\to B:$

Conditions: $\bf 1)$ $F(\emptyset)=\emptyset;$

$\bf 2)$ If $(X_i)_{i\in I}$ is a family of sets in $\mathrm{P}(A),$ then $F\bigg(\displaystyle\bigcup_{i\in I}X_i\bigg)=\bigcup_{i\in I} f(X_i);$

$\bf 3)$ If $X\in \mathrm{P}(A)$ is a singleton, then $F(X)\in \mathrm{P}(B)$ is also a singelton.

We have the following theroem:

Theorem 1: If map $F:\mathrm{P}(A)\to \mathrm{P}(B)$ satisfies all three conditions above, then there exists an $\pmb{ unique}$ map $f:A\to B,$ such that $F=f^{\to}.$

No. Define F(X) = F for all X subset E.
The reader is expected to distinguish which F, F is.
Assume f:E -> Y and let f' be the induced map.
Let x be a point in E. Then f'({x}) is a singleton.
So F /= f' if F is multipoint.
Again be sure to distinguish painday's double useage of F.

A simpler counter example is for F to map
every subset of E to the empty set.

• Thanks for pointing out, and I have corrected the misuse of symbols now. – painday Apr 16 '18 at 3:22

Proof of the theorem 1: Since $F$ satisfies the last condition, now we can define a relation $f$ on set $A\times B$ as $$f:=\left\{(x,y)|\ x\in A,\ y\in F(\left\{x\right\})\right\}.$$ We can easily check that $f$ is actually a map between $A$ and $B,$ since $F(\left\{x\right\})$ is a singleton.

Now take an arbitrary set $X\in \mathrm{P}(A),$ then if $X=\emptyset,$ then according the the first condition, we have $F(X)=\emptyset=f^{\to}(X).$ If $X\neq\emptyset,$ then applying the second condition and the third condition we obtain that $$F(X)=\displaystyle\bigcup_{x\in X} F(\left\{x\right\})=\bigcup_{x\in X} f^{\to}(\left\{x\right\})=f^{\to}(X).$$

Now we see that $F=f^{\to}.$ It's easy to check that the map $f$ is unique.

Q.E.D.

Theorem 2: Let $F:\mathrm{P}(B)\to \mathrm{P}(A)$ be a map, and if $F$ satisfies the conditions below, then there exists some map $f:A\to B,$ such that $F=f^{\leftarrow}.$

$\bf 1)$ $F(B)=A;$

$\bf 2)$ $x,y\in B,\ x\neq y\Longrightarrow F(\left\{x\right\})\cap F(\left\{y\right\})=\emptyset;$

$\bf 3)$ If $(X_i)_{i\in I}$ is a family of sets in $\mathrm{P}(B),$ then $\displaystyle F\bigg(\bigcup_{i\in I}X_i\bigg)=\bigcup_{i\in I} F(X_i).$

Proof: First we define the set $W:=\left\{x\in B|\ F(\left\{x\right\})\neq\emptyset\right\},$ and $V:=\left\{F(\left\{x\right\}|\ x\in W\right\},$ then applying to the first and the last condition we obtain the relation $$A=\displaystyle\bigcup_{x\in B}F(\left\{x\right\})=\bigcup_{x\in W}F(\left\{x\right\})=\bigcup V.$$ Now define a relation on set $A\times Z$ to be $$p:=\left\{(x,y)|\ x\in A,\ y\in Z,\ x\in y\right\},$$ then we can use the second condition to check that $p$ is actually a map between $A$ and $Z.$

Define a map $g:Z\to B,\ F(\left\{x\right\})\to x,$ then applying the second condition we see that $g$ is well-defined.

Finally, let $f:=g\circ p,$ then we can easily check that $F=f^{\leftarrow}.$

Q.E.D.