Question

Write a formal proof where you define a function to prove that the sets 3Z and 6Z have the same cardinality, meaning |3 Z| = |6 Z|. 3Z = {..., −3, 0, 3, 6, 9, ...} and 6Z = {..., −6, 0, 6, 12, 18, ...}

Z = integers

Answer 1

Consider the function defined by

We will show that this is a bijection and thus it will follow that  

Let us assume that  

Hence, is injective.

For any , we have  

Note that

since

Thus, is surjective.

Hence, is bijective and the result follows.