(adsbygoogle = window.adsbygoogle || []).push({}); Dilema about Z_10--->Z_10??

What if somebody asked you to show that the following is a homomorphism

[tex]\theta:Z_{10}->Z_{10}, \theta(\bar m)=5\bar m[/tex]

I already know how to show this one in an elegant manner, it is not difficult at all. But i am wondering if you would chose a hard way to prove it, like i am gonna describe the following, what do u think is that correct?

[tex]Z_{10}=\{\bar 0,\bar 1,\bar 2,\bar 3,\bar 4,\bar 5,\bar 6,\bar 7,\bar 8,\bar 9\}[/tex]

THe way i choose to do this is by first observing that for any element [tex]\bar a \in Z_{10}[/tex] if a=2k then [tex]\theta(\bar a)=\bar 0[/tex] if a=2k+1 then [tex]\theta(\bar a)=\bar 5[/tex]

So. in order to prove that it preserves addition, we would take the following cases.

Let [tex]\bar a, \bar b \in Z_{10}[/tex]

1. a=2k, b=2k

2.a=2k, b=2k+1

3.a=2k+1, b=2k+1

Do the same to show that it preserves the multiplication. And show that in all these cases it preserves both operation.

SO, i know that this might not be elegant at all, but how would you mark it (a) Right or (b) Wrong???

I would really appreciate your opinion on this.

Many thanks1

P.S Me and my friend are having an argument about this, he says that this doesn't prove the point at all, while i think the contrary.

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# Dilema about Z_10->Z_10?

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