In order for a proof by mathematical induction to be
Chapter 5, Problem 30E(choose chapter or problem)
Problem 30E
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 Find the mistakes in the “proofs” by mathematical induction.
Exercise
“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 = {a1, a2, ..., ak, ak+1} be any set of k + 1 numbers. Form two subsets each of size k:
B = {a1, a2, a3, ..., ak} and
C ={a1, a3, a4, ..., ak+1}.
(B consists of all the numbers in A except ak+1, and C consists of all the numbers in A except a2.) By inductive hypothesis, all the numbers in B equal a1 and all the numbers in C equal a1 (since both sets have only k numbers). But every number in A is in B or C, so all the numbers in A equal a1; hence all are equal to each other.”
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