Let n be a positive integer. Define f : Z ! Zn by f(a) = [a]. Is f a homomorphism? One-to-one? Onto? An isomorphism? Explain.
Can any one help? Thanks very much.
Since [ ] is the greatest integer function we have,Originally Posted by suedenation
for all integers.
1)Onto: We need to show that for all we can find such a such as, which is true if you take .
2)One-to-One: We need to show that if then, . Based on the first paragraph that are integers, we have that
3)Homomorphism: We need to show that,
Because, are integers we have,
. Which is definitely not true.
Thus, this map is not an isomorphism. Think about it, how can you have an isomorphsim between a finite and an infinite set? Impossible.
Well, the method is the same as ThePerfectHacker has demonstrated.
For the homomorphism part, check whether is equal to Not hard at all
Ifr were to be one-to-one, you would have had to show that Try testing this, with integers like and .
The onto part is also not hard. Just consider an equivalence class , and find an integer (obvious!) to map to this.
And, since an isomorphism needs to be one-to-one, this (is?/is not?) an isomorphism.