Discrete Math Define F Z Z by the rule Fn 2 3n for all int

Discrete Math

Define F : Z Z by the rule F(n) = 2 3n, for all integers n. Is F one-to-one? Prove or give a counterexample. Is F onto? Prove or give a counterexample.

Solution

Let, m,n so that

f(m)=f(n)

2-3n=2-3m

HEnce

3n=3m

Hence, n=m

So f is injective

F is not onto because: 2-3n = 2 mod 3

So all integers in image of F are 2 mod 3

HEnce F is not onto.