Thread: Showing that the powerset of odd numbers is uncountable

I have already proved that the powerset of the natural numbers is uncountable. To show that P(M) is uncountable where M={n∈N | 2 divides n) I think I need to construct a bijection between P(N) and P(M)? I'm unsure of exactly how to achieve this - I can only imagine a bijection between N and M.

Any help would be appreciated.

Originally Posted by StaryNight

I have already proved that the powerset of the natural numbers is uncountable. To show that P(M) is uncountable where M={n∈N | 2 divides n) I think I need to construct a bijection between P(N) and P(M)? I'm unsure of exactly how to achieve this - I can only imagine a bijection between N and M.

This is a well-known result: If , equipotent, then .
After all there is a bijection .
So result follows that subsets of biject to subsets of .

Let f be a bijection between M and N.

Define g:PM -> PN by

g(x) = the f-image of x.

It's trivial to verify that g is a bijection onto PN.