In my textbook are three interesting properties listed (which I would like to prove)
(1) A is recursively enumerable iff A is the domain of a partial computable function
(2) A is recursievly enumerable iff A is the range of a partial coimputable function
(3) If $f$ is a monoton total computable function => range is recurisvely enumerable and recursiv
Idea for (2) First I can assume $A\not=\emptyset$ and let $x'\in A$. Now I want to find a program $\phi$ such that $f(x)=\phi(x,a)=x'$ for a big enough $a$.
Idea for (3): I think it has something to do with the Church Turing thesisbut do not know how to write it down formally.
Idea for (1): In my eyes this is also intuitively clear because the program need to list every element from the domain to start the calculations but I do not know how to write down.
I would be very thankful if you can help me with this three points.
Generally, (1) is the definition of recursively enumerable, so pending your providing the definition you're using, I won't address that one. Moreover, I'll assume (1) in order to prove (2) and (3).
For (2), you want a single function that outputs all (and only) elements of $A$. So if the domain $\varphi$ is $A$, then you can write another program, $\psi$ that simulates $\varphi$ but outputs only the elements on which $\varphi$ halts.
For (3), since the range of $f$ is r.e., we only need to show that the complement of its range is r.e. (since a set is recursive iff it and its complement are r.e.). To enumerate elements into the complement just wait until larger elements enter the range of $f$.