Let 3 Z= { n ∈ Z | n = 3 k , for some integer  k } .

Prove that Z and 3 Z have the same cardinality.

To prove:

Z and 3Z have the same cardinality.

Given information:

Let 3X={n∈Z|n=3k,for some integer k}.

Concept used:

A function is said to be one-to-one function if distinct elements in domain must be mapped with distinct elements in codomain.

A function is onto function if each element in codomain is mapped with at least one element in domain.

Proof:

For this define a function f:Z→3Z as f(n)=3n.

Objective is to show that f is one to one and onto