Question

Let f : Z12 ? Z4 by f([a]12) = [3a]4. (a) Prove that f is a well defined function. (b) Prove that f is not a ring homomorphism. (c) Is f one-to-one? Is f onto? Prove either one of your choice.

          Let f : Z12 ? Z4 by f([a]12) = [3a]4.

(a) Prove that f is a well defined function.
(b) Prove that f is not a ring homomorphism.
(c) Is f one-to-one? Is f onto? Prove either one of your choice.

Added by Marta S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/02/2023

Step 1

So, assume that [a]12 = [b]12. This means that a and b differ by a multiple of 12, i.e., a - b = 12k for some integer k. Now, we have f([a]12) = [3a]4 and f([b]12) = [3b]4. We need to show that [3a]4 = [3b]4. Since [a]12 = [b]12, we have a = b + 12k. Show more…

Show all steps

Thanks for your feedback!

Let f : Z12 → Z4 by f([a]12) = [3a]4. (a) Prove that f is a well defined function. (b) Prove that f is not a ring homomorphism. (c) Is f one-to-one? Is f onto? Prove either one of your choice.