Checking if f is an isomorphism: did I prove it correctly?

[Joe20]

Hi, I have attached the question and the solutions to part a and b of this question. Would like someone to verify if I have done anything wrong. Greatly appreciate it! Thanks.

Would also like to check if there is a simpler method to prove f is an isomorphism? Thanks

 

[Euge]

Hi Alexis87,

Your proof for part (a) is somewhat circular — the goal was to prove $[a]_6 = _6$ implies $([a]_2,[a]_3) = (_2, _3)$, but you started with that claim. Remove it.

For part (b), remove the beginning part of your proof. To show that $f$ is bijective, you could show that the images $f(j)$, $j = 0,\ldots, 5$, are distinct, and so $f$ is injective. Since $f$ is an injective mapping between two sets of the same cardinality, it is also surjective. Hence, $f$ is bijective.