Questions: Introduction to Direct Proof

Question 1

To prove 'If n is an odd integer, then n² is odd' by direct proof, what should you assume?

• A. Assume n² is odd
• B. Assume n is an odd integer
• C. Assume n² is even
• D. Assume n is an even integer

Answer: (B)

In a direct proof of 'If P, then Q,' you assume P (the hypothesis) and derive Q (the conclusion). Here P is 'n is an odd integer,' so you assume that and work toward showing n² is odd. Starting from the conclusion (option A) would be working backwards; starting from the negation (options C, D) would be a different proof strategy.

Question 2

A direct proof of a statement about most even numbers can be completed by checking the first 100 even numbers.

• T. True
• F. False

Answer: False

Checking specific cases is testing, not proving. A direct proof must work for ALL even numbers simultaneously. You achieve this by using the definition (an even number is 2k for some integer k) and reasoning with the variable k, which represents any integer. The proof then covers every even number at once, not just the ones you checked.

Question 3

Write a direct proof that the product of two odd numbers is odd.

Model answer: Let a and b be odd. By definition, a = 2j + 1 and b = 2k + 1 for some integers j and k. Then ab = (2j+1)(2k+1) = 4jk + 2j + 2k + 1 = 2(2jk + j + k) + 1. Since 2jk + j + k is an integer, ab has the form 2m + 1, so ab is odd.

The proof follows the direct proof template: assume the hypothesis (a and b are odd), unpack the definition (write each as 2·integer + 1), perform algebra, and show the result matches the definition of the conclusion (the product has the form 2·integer + 1, hence odd). Every step is justified and no cases are left uncovered.