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Prove that P(N) is equipotent with the set of functions 2N = {f : N → {0, 1} : f is a function}. In particular, the cardinality of P(N) is 2ℵ

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.5: Permutations And Inverses

Problem 6E: 6. Prove that if is a permutation on , then is a permutation on .

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Question

Prove that P(N) is equipotent with the set of functions

2N = {f : N → {0, 1} : f is a function}.

In particular, the cardinality of P(N) is 2ℵ0

Expert Solution

Step 1

Given that,

$\{f : N \to {0, 1} : f i s a f u n c t i o n\}$

The aim is to show that, $P (N) = \{f : N \to {0, 1} : f i s a f u n c t i o n\}$

The infinite sequences of 0s and 1s can be stated as a result about the power set of N, P(N).

An infinite sequence is a function f : N → {0, 1}

And the set of such sequences is usually denoted ${\{0, 1\}}^{N}$ .

A subset A ⊂ N defines a function f : N → {0, 1}:

$f (n) = 1$ if $n \in A$ and $f (n) = 0$ if $n \notin A$ .

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