Practice: Proof by Contradiction

A student has tried to prove the following statement by contradiction: β€œFor all even integers π‘š and 𝑛, their product π‘šπ‘› is divisible by 4.” The student’s working is shown below.

  1. Assumption: All integers 𝑛 such that π‘›οŠ¨ is odd are even.
  2. 𝑛 is even; so, write 𝑛=2π‘˜
  3. Then, 𝑛=(2π‘˜)=4π‘˜=2ο€Ή2π‘˜ο…οŠ¨οŠ¨οŠ¨οŠ¨, so π‘›οŠ¨ is even.
  4. This contradicts the assumption that π‘›οŠ¨ is odd.
  5. Therefore, if π‘›οŠ¨ is odd, then 𝑛 must be odd.

Identify the error in their working.

  • AIn the final sentence, they have drawn the wrong conclusion.
  • BIn the second sentence, they should have written π‘š=2π‘˜+1 and 𝑛=2𝑙+1.
  • CIn the first sentence, they have given the incorrect negation of the statement they want to prove.
  • DIn the fourth sentence, they have stated the wrong assumption.
  • EIn the third sentence, the algebra is incorrect.

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