Question: Is f: Z2 -> Z2 with f(x) = -x a homomorphism?

[chaotixmonjuish]

(A) f: Z -> Z with a function defined as f(x)=-x

1.) f(0)=0
2.) f(a)-f(b)=-a+b=-a+b=f(a-b)
3.) -f(a)=a=a=f(-a)

Conclusion: a homomorphism

(B) f: Z₂ -> Z₂ with a function defined as f(x)=-x
This one I'm not sure, I doubt it is a homomorphism though.

f: g -> g

 

[Dick]

Z_s? s equal what? s=2 certainly is a homomorphism. s>2 may cause problems. But not with f(0).

 

[chaotixmonjuish]

Sorry, I didn't even realize all the things i left off. But anyway, are my three proofs correct?

 

[Dick]

What three proofs? I thought there were maximum two problems here. f(x)=-x for i) Z->Z and ii) Z_2->Z_s. Whatever s is. Homomorphism, yes or no. You are being super unclear.

 

[chaotixmonjuish]

Wow, sorry. I just transposed everything from my notes quickly. The S was suppose to be a 2, which I fixed. I also only did put up two proofs just to see if Z -> Z and Z2 -> Z2 were homomorphisms. I arrived at the same conclusion for both. I actually didn't type out the third one, I'm not sure why.