10) Identify the mistake in the following “proof" by mathematical induction. (Fake) Theorem: For any positive integer n, 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 induction step, let A = {a1, a2, az, ..., an+1} be any set of n +1 numbers. Form two subsets each of size n: B = {q1,a2, az,...,a,} and C = {q1,a3, as, … . .,an+1}. (Precisely, B consists of all the numbers in A except an+1; and C consists of all the numbers in A except a2.) By the induction hypothesis, all the numbers in B equal a1 and all the numbers in C equal a1, since both sets have only n numbers. But every number in A is in B or C, so all the numbers in A equal a1; thus all are equal to each other.

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.