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## Homework Statement

Show that |P(N)|=|R|. (R=reals, |X| is the cardinality of X).

## Homework Equations

## The Attempt at a Solution

#1.) P(N) --> R

Given any element, A, of P(N), construct a decimal expansion .x1x2x3x4... by the rule that x_i=1 (if i is in A) and x_i=0 (if i is not in A).

So the element {1,7} would give .1000001

This map is 1-1 but not onto the Reals.

#2.) R --> P(N)

If I can show this direction then #3 follows. I know a little about the mapping of [0,1] onto the Reals. I cannot determine if that helps here.

#3.) Using the Cantor-Schroeder-Bernstein Theorem, |P(N)|=|R|.