Z+ and Z are well-ordered. That means every subset has a least element. There's your choice function. Take the least element of each set.
Find, if possible, a choice function for each of the following collections without using the axiom of choice.
(a) The collection A of nonempty subsets of Z+
(b) The collection B of nonempty subsets of Z
There are a (c) and (d) as well, but I am hoping that help with these first two will mean I can do the last two.
I do not know how to do this. If someone could explain the process and also give a little more in depth explanation of a choice function, it would really help me.