# Showing that the powerset of odd numbers is uncountable

• Apr 20th 2011, 12:02 PM
StaryNight
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.
• Apr 20th 2011, 12:19 PM
Plato
Quote:

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 $A\sim B$, equipotent, then http://quicklatex.com/cache3/ql_5da9...68b5a03_l3.png.
After all there is a bijection http://quicklatex.com/cache3/ql_cfdb...b9342b1_l3.png.
So result follows that subsets of $A$ biject to subsets of $B$.
• Apr 22nd 2011, 08:05 AM
MoeBlee
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.