Let f Z3 rightarrow Z3 be the function given by fx x2 2 mo

Let f: Z_3 rightarrow Z_3 be the function given by f(x) = x^2 + 2 (mod 3). Is f is an injection? Is f a surjection? Is f a bijection? Briefly explain your conclusions.

Solution

f(1)=1^2+3=4 mod 3

f(-1)=(-1)^2+3=4 mod 3

Hence, f is not an injection

x^2 can take only values :0,1 mod 3

HEnce, x^2+2 can take only values:2,3 ie 0,2 mod 3

HEnce ,f is not a surjection

f is neither injection nor surjection hence not a bijection