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## Homework Statement

Using the definitions, prove that the set of odd integers is countably infinite.

## Homework Equations

Definition: The set A is countably infinite if its elements can be put in a 1-1 correspondence with the set of positive integers.

## The Attempt at a Solution

I am trying to think of a function that maps the positive integers into the odd integers.

The function I am coming up with is piecewise defined.

I am unsure how to type it, but...

f(n) =

n-2 when n is odd and n>2

-n-1 when n is even

-n when n=1

Testing some values, I got this

f(1)=-1

f(2)=-3

f(3)=1

f(4)=-5

f(5)=3

It looks like this works, but I am unsure how to show a piecewise defined function is 1-1 and onto. I think I am missing something.