Name ______________________________
Grade the following proof. Assign an “A” if the statement and proof are correct. Assign a “C” if the statement is correct but the proof is not and assign an “F” if the statement and proof are incorrect. Justify a grade other than “A”.
Claim: For every natural number n, n2 + n is odd.
“Proof.” The number n = 1 is odd. Suppose n Î ℕ and n2 + n is odd. Then
(n+1)2 + (n+1) = n2 + 2n + 1 + n + 1 = (n2 + n) + (2n + 2)
is the sum of an odd and an even number. Therefore, (n+1)2+ (n+1) is odd. By the PMI, the property that n2 + n is odd is true for all natural numbers n.
Solution: F. The statement is false, consider the case n = 1, in this case n2 + n = 2 which is not odd. This also shows where the error in the “proof” is ... the base case is treated incorrectly.