Define f: Z/4Z → Z/4Z by f([a]) = [3a+1]

Question

I need to show this function is well defined

For well defined, I was thinking something along the lines of:

Assume [a1] = [a2] in Z/4Z. Then, a1 is congruent to a2(mod4). So, 4 | a1 - a2. Thus, 4 | 3(a1 - a2) by property of divisibility. Then i can just say 4| 3a1 - 3a2 + 1 -1 which is the same thing as 4| (3a1 +1) - (3a2 +1).

Right?

How about surjective/injective?

Answer 1

You need to show $4|[(3a_1+1)-(3a_2+1)]$. By your work, you're almost there.