   Chapter 7.2, Problem 56ES ### 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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# In Example 7.2.8 a one-to-one correspondence was de?ned from the power set of {a, b} to the set of all strings of 0’s and 1’s that have length 2. Thus the elements of these two sets can be matched up exactly, and so the two sets have the same number of elements. a. Let X = { x 1 ,   x 2 ,   .   .   .   , x n } be a set with n elements. Use Example 7.2.8 as a model to define a one-te-one correspondence from P(X), the set of all subsets of X, to the set of a1l strings of 0’s and 1’s that have length n.b. In Section 9.2 we show that there are 2 n strings of 0’s and 1’s that have length n. What does this allow you to conclude about the number of subsets of P(X)? (This provides an alternative proof of Theorem 6.3.1.)

To determine

(a)

To prove:

If X={x1,x2,......,xn} be a set with n elements, then a one-to-one correspondence from P(X), the set of all subsets of X, to the set of all strings of 0's and 1's that have length n.

Explanation

Given information:

X={x1,x2,.......,xn} be a set with n elements.

Concept used:

In one-to-one function, distinct elements in domain are mapped with distinct elements in co-domain.

Proof:

Let X={x1,x2,.......,xn} and S be the set of all strings of 0's and 1's of length n.

Define f:P(X)S such that

f(A)=a string of 0's and 1's of length n such that xiA if and only if ith place is 1 in the string.

To show f is one-to-one, let f(A)=f(B).

Let xiA

ith place is 1 in the string f(A)

To determine

(b)

To find:

Total number of subsets of a set with n elements.

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