Give an explicit definition for the different homomorphisms $\psi_1 ,\psi_2:\mathbb{Z}_{5}\to\mathbb{Z}_{5}$. Can anyone please help me define clearly using the definition for a group homomorphism that for the given two homomorphisms it holds for all $g_1,g_2\in G$? 
We know that the definition gives that there exists two groups, $G$ and $H$ and a function  $\psi_1:G\to H$. This is then called a homomorphism if the equality $\psi_1(g_1+g_2)=\psi_1(g_1)+\psi_1(g_2)$ holds for all $g_1,g_2\in G$. 
So as a start,we take the first function $\psi_1:\mathbb{Z}_{5}\to\mathbb{Z}_{5}$. If we let $x,y\in\mathbb{Z}_{5}$, then $\psi_1(x+y)=\psi_1(x)+\psi_1(y)$?
Please brief me out on how to do this problem.

The group $\mathbb{Z}_5$ is written additively, so the homomorphism property spells as $\psi_1(x + y) = \psi_1(x) + \psi_1(y)$ in this case.
There are several ways to construct such a function. To find them all, try the following:
1) Which possible choices are there for $\psi_1(1)$?
2) Assure yourself, that by the homomorphism property all values $\psi_1(x)$ are determined once that a choice for $\psi_1(1)$ was made.
