   Chapter 7.4, Problem 34ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Let P ( s ) be the set of all subsets of set S, and let T be the set of all functions from S to { 0 , 1 } . Show that P ( S ) and T have the same cardinality.

To determine

To prove:

P(S) and T have the same cardinality where P(S) be the set of all subsets of set S and let T be the set of all functions from S to {0,1}.

Explanation

Given information:

P(S) be the set of all subsets of set S and let T be the set of all functions from S to {0,1}.

Concept used:

P(S) be the power set.

Proof:

Consider a set S and its power set that is set of all subsets, is P(S).

Let T be the set of all functions begin from S to {0,1}.

The objective is to show that the power set P(S) and T have the same cardinality.

Consider P(S) be the set of all subsets of set S, let T be the set of all functions from S to {0,1}.

Consider a function f defined as follows.

f:P(S)T

For each subset A of S ,

Let f(A)=xA is the characteristic function of A.

The function xA:S{0,1} is defined as follows.

xA(x)={1if xA0  if xA for all xS

Now, to show that f is a bijection function.

It is enough to prove that xA is a bijection

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