Define f Z rightarrow Z by the rule fn 2n for all integers

Define f: Z rightarrow Z by the rule f(n) = 2n, for all integers n. (i) Is f one-to-one? Prove or give a counterexample. (ii) Is f onto? Prove or give a counterexample. b. Let 2Z denote the set of all even integers. That is, 2Z = {n Z | n = 2k, for some integer k}. Define h: Z rightarrow 2Z by the rule h(n) = 2n, for all integers n. Is h onto? Prove or give a counterexample.

Solution

b.

Let, k be in 2Z hence ,k=2m for some integer m

Hence

h(m)=2m=k

But k was arbitrary element in 2Z and we have proved for each k in 2Z there is an m in Z so that

h(m)=k

Hence, h is onto.