   Chapter 6.3, Problem 26ES ### 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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# The following problem, devised by Ginger Bolton, apeared in the January 1989 issue of the College Mathematics Journal (Vol.20, No. 1, p.68); Given a positive integer n ≥ 2 , let S be the set of all nonempty subets of { 2 , 3 , ... , n } . For each S i ∈ S , let P i be the product of the elements of S i Prove or disprove tht

To determine

Prove or disprove that

i=12n11P1=(n+1)!21

Explanation

Given information The following problem, devised by Ginger Bolton, appeared in the January 1989 issue of the college Mathematics journal (Vol.20,No.1, p. 68): Given a positive integer n2, let S be the set of all nonempty subsets of {2,3,......,n}. For each SiS, let Pi be the product of the elements of Si.

Concept used:

Principle of mathematical induction is used to prove mathematical terms.

Calculation:

Let S be the set of all nonempty subsets of {2,3,.....,n} for any positive integer n2.

Write the following statement P(n) :

For each SiS let p1 be the product of the elements of Si such that summation of product is given by the formula i=12n11P1=(n+1)!21.

For n=2,S={{2}} and there is only single element of S, namely S1={2}.

Then P1=2

That is, the product of all nonempty subsets of {2} is 2.

Also

i=11p1=( n+1)!21=31=2

Thus, P(2) is true.

Now, prove that for all integers k2, it P(k) is true, then P(k+1) is also true.

Therefore, in the inductive step, consider the set of all nonempty subsets of {2,3,.....,k}

Now, for P(k+1), consider the set of all nonempty subsets of {2,3,.....,k+1}.

Now, any subset of {2,3,.....,k+1} either contains the element k+1 or does not contain the element k+1

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