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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

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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Sect-7.1 P-2ESSect-7.1 P-3ESSect-7.1 P-4ESSect-7.1 P-5ESSect-7.1 P-6ESSect-7.1 P-7ESSect-7.1 P-8ESSect-7.1 P-9ESSect-7.1 P-10ESSect-7.1 P-11ESSect-7.1 P-12ESSect-7.1 P-13ESSect-7.1 P-14ESSect-7.1 P-15ESSect-7.1 P-16ESSect-7.1 P-17ESSect-7.1 P-18ESSect-7.1 P-19ESSect-7.1 P-20ESSect-7.1 P-21ESSect-7.1 P-22ESSect-7.1 P-23ESSect-7.1 P-24ESSect-7.1 P-25ESSect-7.1 P-26ESSect-7.1 P-27ESSect-7.1 P-28ESSect-7.1 P-29ESSect-7.1 P-30ESSect-7.1 P-31ESSect-7.1 P-32ESSect-7.1 P-33ESSect-7.1 P-34ESSect-7.1 P-35ESSect-7.1 P-36ESSect-7.1 P-37ESSect-7.1 P-38ESSect-7.1 P-39ESSect-7.1 P-40ESSect-7.1 P-41ESSect-7.1 P-42ESSect-7.1 P-43ESSect-7.1 P-44ESSect-7.1 P-45ESSect-7.1 P-46ESSect-7.1 P-47ESSect-7.1 P-48ESSect-7.1 P-49ESSect-7.1 P-50ESSect-7.1 P-51ESSect-7.1 P-52ESSect-7.1 P-53ESSect-7.2 P-1TYSect-7.2 P-2TYSect-7.2 P-3TYSect-7.2 P-4TYSect-7.2 P-5TYSect-7.2 P-6TYSect-7.2 P-7TYSect-7.2 P-8TYSect-7.2 P-9TYSect-7.2 P-10TYSect-7.2 P-11TYSect-7.2 P-1ESSect-7.2 P-2ESSect-7.2 P-3ESSect-7.2 P-4ESSect-7.2 P-5ESSect-7.2 P-6ESSect-7.2 P-7ESSect-7.2 P-8ESSect-7.2 P-9ESSect-7.2 P-10ESSect-7.2 P-11ESSect-7.2 P-12ESSect-7.2 P-13ESSect-7.2 P-14ESSect-7.2 P-15ESSect-7.2 P-16ESSect-7.2 P-17ESSect-7.2 P-18ESSect-7.2 P-19ESSect-7.2 P-20ESSect-7.2 P-21ESSect-7.2 P-22ESSect-7.2 P-23ESSect-7.2 P-24ESSect-7.2 P-25ESSect-7.2 P-26ESSect-7.2 P-27ESSect-7.2 P-28ESSect-7.2 P-29ESSect-7.2 P-30ESSect-7.2 P-31ESSect-7.2 P-32ESSect-7.2 P-33ESSect-7.2 P-34ESSect-7.2 P-35ESSect-7.2 P-36ESSect-7.2 P-37ESSect-7.2 P-38ESSect-7.2 P-39ESSect-7.2 P-40ESSect-7.2 P-41ESSect-7.2 P-42ESSect-7.2 P-43ESSect-7.2 P-44ESSect-7.2 P-45ESSect-7.2 P-46ESSect-7.2 P-47ESSect-7.2 P-48ESSect-7.2 P-49ESSect-7.2 P-50ESSect-7.2 P-51ESSect-7.2 P-52ESSect-7.2 P-53ESSect-7.2 P-54ESSect-7.2 P-55ESSect-7.2 P-56ESSect-7.2 P-57ESSect-7.2 P-58ESSect-7.3 P-1TYSect-7.3 P-2TYSect-7.3 P-3TYSect-7.3 P-4TYSect-7.3 P-5TYSect-7.3 P-1ESSect-7.3 P-2ESSect-7.3 P-3ESSect-7.3 P-4ESSect-7.3 P-5ESSect-7.3 P-6ESSect-7.3 P-7ESSect-7.3 P-8ESSect-7.3 P-9ESSect-7.3 P-10ESSect-7.3 P-11ESSect-7.3 P-12ESSect-7.3 P-13ESSect-7.3 P-14ESSect-7.3 P-15ESSect-7.3 P-16ESSect-7.3 P-17ESSect-7.3 P-18ESSect-7.3 P-19ESSect-7.3 P-20ESSect-7.3 P-21ESSect-7.3 P-22ESSect-7.3 P-23ESSect-7.3 P-24ESSect-7.3 P-25ESSect-7.3 P-26ESSect-7.3 P-27ESSect-7.3 P-28ESSect-7.3 P-29ESSect-7.3 P-30ESSect-7.4 P-1TYSect-7.4 P-2TYSect-7.4 P-3TYSect-7.4 P-4TYSect-7.4 P-5TYSect-7.4 P-6TYSect-7.4 P-7TYSect-7.4 P-8TYSect-7.4 P-9TYSect-7.4 P-1ESSect-7.4 P-2ESSect-7.4 P-3ESSect-7.4 P-4ESSect-7.4 P-5ESSect-7.4 P-6ESSect-7.4 P-7ESSect-7.4 P-8ESSect-7.4 P-9ESSect-7.4 P-10ESSect-7.4 P-11ESSect-7.4 P-12ESSect-7.4 P-13ESSect-7.4 P-14ESSect-7.4 P-15ESSect-7.4 P-16ESSect-7.4 P-17ESSect-7.4 P-18ESSect-7.4 P-19ESSect-7.4 P-20ESSect-7.4 P-21ESSect-7.4 P-22ESSect-7.4 P-23ESSect-7.4 P-24ESSect-7.4 P-25ESSect-7.4 P-26ESSect-7.4 P-27ESSect-7.4 P-28ESSect-7.4 P-29ESSect-7.4 P-30ESSect-7.4 P-31ESSect-7.4 P-32ESSect-7.4 P-33ESSect-7.4 P-34ESSect-7.4 P-35ESSect-7.4 P-36ESSect-7.4 P-37ESSect-7.4 P-38ES