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 n ≥ 2 , let S be the set of all nonempty subsets of { 2 , 3 , ...... , n } . For each S i ∈ S , let P i be the product of the elements of S i . 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 n ≥ 2 . Write the following statement P ( n ) : For each S i ∈ S let p 1 be the product of the elements of S i such that summation of product is given by the formula ∑ i = 1 2 n − 1 − 1 P 1 = ( n + 1 ) ! 2 − 1 . For n = 2 , S = { { 2 } } and there is only single element of S , namely S 1 = { 2 } . Then P 1 = 2 That is, the product of all nonempty subsets of { 2 } is 2 . Also ∑ i = 1 1 p 1 = ( n + 1 ) ! 2 − 1 = 3 − 1 = 2 Thus, P ( 2 ) is true. Now, prove that for all integers k ≥ 2 , 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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ISBN: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

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Chapter 6.3, Problem 26ES

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Prove or disprove that

∑i=12n−1−1P1=(n+1)!2−1

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Chapter 6.3, Problem 25ES

Chapter 6.3, Problem 27ES

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Prove or disprove that ∑ i = 1 2 n − 1 − 1 P 1 = ( n + 1 ) ! 2 − 1

Solution Summary: The author explains how the principle of mathematical induction is used to prove mathematical terms.

Concept explainers

Prove or disprove that

∑i=12n−1−1P1=(n+1)!2−1

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Chapter 6 Solutions

Prove or disprove that ∑ i = 1 2 n − 1 − 1 P 1 = ( n + 1 ) ! 2 − 1

Solution Summary: The author explains how the principle of mathematical induction is used to prove mathematical terms.

Concept explainers

Prove or disprove that ∑i=12n−1−1P1=(n+1)!2−1

Want to see the full answer?

Chapter 6 Solutions

Prove or disprove that

∑i=12n−1−1P1=(n+1)!2−1