Practice question · Select all that apply
The claim is: for every integer n, if n squared is odd then n is odd. Select every approach that would genuinely establish it.
Hints
- Three of the standard techniques apply here; work out which statement each option is actually assuming.
- One option accumulates examples and one option would refute the claim rather than prove it.
Show the answer
- A. Assume n squared is odd, write n squared = 2k + 1, and reason directly to n being odd
- C. Assume n is even and show that n squared is even
- D. Assume n squared is odd and n is even, and derive a contradiction
Why
Option 1 is the contrapositive, option 4 is contradiction, option 1 is a direct proof, all three are valid routes. Checking particular values proves nothing about a universal claim, and a counterexample would disprove it, not prove it. Notice how naturally the contrapositive falls out here: 'if n is even then n squared is even' is a one-line argument.
Practise Proof Techniques
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