1. The problem statement, all variables and given/known data Using the definitions, prove that the set of odd integers is countably infinite. 2. Relevant 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. 3. 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.