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 operation in ${\bf Z}_5$ is addition, so what you want is $$\psi(x+y)=\psi(x)+\psi(y)$$ 
You should 


*

*prove that $\psi(0)=0$, 

*prove that if you know $\psi(1)$ then you can work out $\psi(m)$ for all $m$, 

*see what happens when you take different values for $\psi(1)$. 
