   Chapter 6.3, Problem 25ES ### 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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# Use mathematical induction to prove that for every integer n ≥ 2 , if a set S has n elements, then the number of subsets of S with an even number of elements equals number of subsets of S with an odd number of elements.

To determine

To prove that the number of subsets of S with an even number of elements is equal to the number of subsets of S with an odd number of elements by mathematical induction.

Explanation

Given information:

For every integer n2, the set S has n elements.

Formula used:

The number of subsets for a set of n elements is 2n.

Calculation:

Let’s prove the result by mathematical induction.

Basic step:

n=1

The set S contains only one element.

The result is true for a 1-element set.

Inductive step:

Assume that the result is true for k such that 1kn.

Let’s prove that this result is true for k+1-element set.

Let A={1,2,3,...k,k+1} and B={1,2,3,...k}.

That is set A contains k+1, elements and set B contains elements with the exception of the (k+1)th element of set B

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