Writing a Proof Someone Else Can Check · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Writing proofs a stranger can verify

Mathematics · Mathematical Thinking · ages 22-23
Name ______________________   Date ____________

  1. Which opening line announces the structure of the proof?

    • We prove directly that the sum of two odd numbers is even: let a and b be odd.
    • It is obvious that odd numbers add to an even number.
    • This proof was hard to find.
    • Suppose the reader already believes the claim.

  2. Which opening line announces the structure of the proof?

    • It is obvious that odd numbers add to an even number
    • We prove directly that the sum of two odd numbers is even: let a and b be odd
    • This proof was hard to find

  3. If X then Y holds. Which reading is correct?

    • X suffices for Y, and Y is necessary for X
    • X is necessary for Y, and Y suffices for X
    • X and Y are unrelated

  4. Theo begins: we prove by contradiction that root 2 is irrational. Suppose root 2 is rational. Theo has announced the structure in his first lines.

    Circle one:   True   False

  5. Over the integers, which statement is true?

    • For every x there is a y with y > x, for example y = x + 1
    • There is a single y larger than every integer x
    • Both statements say the same thing
    • Neither statement is true

  6. Over the integers, which statement is true?

    • There is a single y larger than every integer x
    • For every x there is a y with y above x, for example y equals x plus 1
    • Both statements say the same thing

  7. A proof uses the hypothesis that n is a prime larger than 2 to conclude n is odd. Where should that hypothesis be named?

    • At the step itself: since n is prime larger than 2, n is odd
    • Only in a preamble, never again
    • Nowhere, since primes are always odd

  8. A proof uses the hypothesis that n is a prime larger than 2 to conclude n is odd. Where should that hypothesis be named?

    • At the step itself: since n is prime larger than 2, n is odd
    • Only in a preamble, never again
    • Nowhere, since primes are always odd
    • In the title of the proof

  9. A proof needs the reading where each student may have solved a different problem. Which rewrite states that order explicitly?

    • There is one single problem that every student solved
    • Some problem was solved by students
    • For each student S there is a possibly different problem P that S solved

  10. A proof needs the reading where each student may have solved a different problem. Which rewrite states that order explicitly?

    • For each student S there is a (possibly different) problem P that S solved
    • There is one single problem that every student solved
    • Some problem was solved by students
    • Students solve problems
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Writing proofs a stranger can verify W1-mt_nZ-zMzj18N-s1

  1. We prove directly that the sum of two odd numbers is even: let a and b be odd. · The first line states the claim and the method (direct proof) and introduces the numbers, so the reader knows what is coming.
  2. We prove directly that the sum of two odd numbers is even: let a and b be odd · It states the claim and the method and introduces the numbers.
  3. X suffices for Y, and Y is necessary for X · The arrow direction fixes sufficiency one way and necessity the other.
  4. True · He stated the claim, the method, and the opening assumption.
  5. For every x there is a y with y > x, for example y = x + 1 · Given any x, y = x + 1 works, but no single y beats every integer.
  6. For every x there is a y with y above x, for example y equals x plus 1 · Given any x, y equals x plus 1 works, but no single y beats every integer.
  7. At the step itself: since n is prime larger than 2, n is odd · Naming the hypothesis where it is used lets the reader check that step.
  8. At the step itself: since n is prime larger than 2, n is odd · Naming the hypothesis where it is used lets the reader check that step on the spot.
  9. For each student S there is a possibly different problem P that S solved · Only the last rewrite pins down the dependence of the problem on the student.
  10. For each student S there is a (possibly different) problem P that S solved · Only the first rewrite pins down the dependence: the problem may vary with the student.
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