Define s Z4 rightarrow Z4 by sa 2a Decide whether or not s

Define s: Z_4 rightarrow Z_4 by s(a) = 2a. Decide whether or not s is 1-1 and onto and prove your answers. Repeat the previous problem, but replace Z_4 with Z_5.

Solution

(7). s(0)=2(0)=0, s(2)=2.2=4=0 (mod4) => s is not one-one. This is not onto as 2a is never going to yield 3 of co-domain.

(8). s(0)=0, s(1)=2 (mod 5), s(2)=4 (mod 5), s(3)=1 (mod 5), s(4)=8=3 (mod 5). This shows that s is one one and onto.