The set of of all integers, positive, negative or zero, is countable. In fact, we can set up the following one-to -one correspondence between and the set of all positive integers. \ \

\ \

More, explicitly, we associate the non-negative integer with the odd number , and the negative integer with the even number ,

\ \

There is a one-to-one correspondence between the set of all integers and the set of positive integers.

\ \

Let and be two sets, which are in one to one correspondence.

For finite sets: A one-to-one correspondence exist if and only if the sets have the same number of elements.

For infinite sets: Two infinite sets of elements have the same transfinite cardinal number if and only if there exists a one-to-one correspondence between the elements of the two sets.

Therefrore, and have same number of elements.

\ \