Show that this is a homomorphism

1. The problem statement, all variables and given/known data
Show that [itex] (\mathbb{Z} _4 , \oplus ) \approx ( \mathbb{Z} ^{0}_{5} [/itex] [itex] , \odot ) [/itex]
Meaning, they are isomorphic. The 0 means the zero is deleted from the set. We are using circle plus and circle dot because we are not allowed to think of the operations as addition and multiplication yet. My trouble is with defining the operation [itex] \theta [/itex].


2. Relevant equations

I know [itex] \theta [/itex] must be operation preserving, that is, [itex] \theta (x \oplus y) = \theta (x) \odot \theta (y) [/itex]

3. The attempt at a solution

I tried defining [itex] \theta [/itex] as [itex] \theta ([x])= [x+1] [/itex]
so, [itex] \theta ([0]) = [1] [/itex] up to [itex] \theta ([3]) = [4] [/itex]
And now, [itex] \theta ([x] \oplus [y]) = \theta ([x \oplus y]) = [x \oplus y +1] [/itex]
Did I make the wrong steps there? I'm not sure how to arrive at [itex] \theta ([x]) \odot \theta ([y]) [/itex]

 

[itex] \theta ([0])=[1] [/itex]
[itex] \theta ([1])=[2] [/itex]
[itex] \theta ([2])=[3] [/itex]
[itex] \theta ([3])=[4] [/itex]

But how do I show it's operation preserving? Do I have to show it using each number, or is there some way to do it using arbitrary elements?

 

Okay, if that's the only way I can certainly do that. thanks for the tip-off about my mapping being wrong. I think with this information I'll be okay here. :)