Theorem: For any integer n 1, all the numbers in a set of n numbers are equal to each

Chapter 5, Problem 30

(choose chapter or problem)

In order for a proof by mathematical induction to be valid, the basis statement must be true for n = a and the argument of the inductive step must be correct for every integer k ≥ a. In 30 and 31 find the mistakes in the “proofs” by mathematical induction.

Theorem:” For any integer n ≥ 1, all the numbers in a set of n numbers are equal to each other. “Proof (by mathematical induction): It is obviously true that all the numbers in a set consisting of just one number are equal to each other, so the basis step is true. For the inductive step, let \(A = {a_{1}, a_{2}, . . . , a_{k} , a_{k+1}}\) be any set of k + 1 numbers. Form two subsets each of size k:

\(B = {a_{1}, a_{2}, a_{3}, . . . , a_{k} }\) and

\(C = {a_{1}, a_{3}, a_{4}, . . . , a_{k+1}}\).

(B consists of all the numbers in A except \(a_{k+1}\), and C consists of all the numbers in A except \(a_{2}\).) By inductive hypothesis, all the numbers in B equal \(a_{1}\) and all the numbers in C equal \(a_{1}\) (since both sets have only k numbers). But every number in A is in B or C, so all the numbers in A equal \(a_{1}\); hence all are equal to each other.”

Text Transcription:

A = {a_1, a_2, . . . , a_k , a_k+1}

B = {a_1, a_2, a_3, . . . , a_k }

C = {a_1, a_3, a_4, . . . , a_k+1}

a_k+1

a_2

a_1