Construct choice functions

Question

Can someone review my solution for the following exercise?

Construct a choice function for

1. All nonempty finite subsets of $\mathbb{R}$

2. All nonempty subsets of $\mathbb{Z}$

Solution:

2.

$A \in \{X: X \subset \mathbb{Z}\} \setminus \{\emptyset\}$

$f(A) = \begin{cases} \min\{x:x\in A\}, & \forall x \in A:x>0 \\ \max\{x:x\in A\}, & \forall x \in A:x<0 \\ 0, & 0 \in A \\ \min\{\vert x \vert: x\in A\}, & \text{else} \end{cases}$

1.

$A \in \{X: X \subset \mathbb{R}, \# A < \infty \} \setminus \{\emptyset\}$

$f(A)=\min\{x:x \in A\}$

Answer 1

The first solution is not entirely correct. If $A=\{-1,2\}$ then $f(A)=1$ as you define it, but $1\notin A$.

Instead, you can compress all the cases into a correctly formed $$f(A)=\min\{x\in A:|x|=\min\{|x|:x\in A\}\}.$$

Answer 2

The definition in 1. is ambiguous. What is $f(\{0,1,2,4\})$? $1$ as the first clause says or $4$ as the second says, or $0$ as the third says?

2. is fine (we could also use the max).