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

5th Edition

ISBN: 9780357097717

Author: EPP

Publisher: CENGAGE L

See similar textbooks

Concept explainers

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”.

Question

Chapter 6.3, Problem 26ES

To determine

Prove or disprove that

∑i=12n−1−1P1=(n+1)!2−1

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.3, Problem 25ES

Chapter 6.3, Problem 27ES

Students have asked these similar questions

Find: lim x →-6 f (x) limx-4 f (x) lim x-1 f (x) lim x →4 f (x) (-6,3) • (-1,5) -8 -7 (-6,-2) 4+ (4,5) (4,2) • (-1,1) -6

3 2 Find: ƒ(1) lim f(x) 14-x 2 ƒ(2) lim f(x) x-2- lim f(x) x+2+ lim f(x) x→4 3 y=f(x)

For each graph below, state whether it represents a function. Graph 1 24y Graph 2 Graph 3 4 2 -8 -6 -4 -2 -2 2 4 6 Function? ○ Yes ○ No ○ Yes ○ No Graph 4 Graph 5 8 Function? Yes No Yes No -2. ○ Yes ○ No Graph 6 4 + 2 4 -8 -6 -4 -2 2 4 6 8 Yes -4++ No

Chapter 6 Solutions

DISCRETE MATHEMATICS WITH APPLICATION (

Ch. 6.1 - The notation is read”______” and means that___Ch. 6.1 - To use an element argument for proving that a set...Ch. 6.1 - Prob. 3TY Ch. 6.1 - An element x is in AB if , and only if,_______Ch. 6.1 - An element x in AB if, and only if,______Ch. 6.1 - An element x is in B-A if, and only if,______Ch. 6.1 - An elements x is in Acif, and only if.______Ch. 6.1 - The empty set is a set with ______Ch. 6.1 - The power set of a set A is _____Ch. 6.1 - Prob. 10TY

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

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 = 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