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. A direct proof uses these lines: let a,b be even; write a=2r,b=2s for integers r,s; then a+b=2(r+s); so a+b is even. Which order lets each line use facts already established?

    • Introduce a,b; add the expressions; choose r,s; conclude evenness.
    • Introduce a,b; choose r,s; add the expressions; conclude evenness.
    • Choose r,s; conclude evenness; introduce a,b; add the expressions.
    • Conclude evenness; introduce a,b; add the expressions; choose r,s.

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

    • Let a and b be odd integers. Their sum is even, and the reader should accept this without a reason.
    • We prove directly that two odd integers have an even sum. Let a and b be odd integers.
    • We found this proof after trying several examples of odd integers and calculating their sums.

  3. From 'if X then Y' alone, which pair of roles is guaranteed?

    • 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. Only Ana and Bo are students, and only P and Q are tasks. Ana solved only P; Bo solved only Q. Compare 'for each student there is a task they solved' with 'there is a task every student solved'. Which verdict fits?

    • The student-first claim holds; the task-first claim fails.
    • The student-first claim fails; the task-first claim holds.
    • Both claims hold because they require the same task for all.
    • Both claims fail because neither student solved both tasks.

  6. Over the integers, which statement is true?

    • There is one fixed integer y that is greater than every integer x.
    • For every integer x there is an integer y with y > x.
    • The two quantifier orders above describe exactly the same requirement.

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

    • At the deduction: since n is prime and n > 2, it is odd, because an even prime is 2.
    • Only at the start of the proof: the reader must infer which later deductions use the prime hypothesis.
    • Nowhere in the proof: calling n prime already rules out every even integer, with no need to exclude 2.

  8. An integer m is divisible by 3. A draft uses this hypothesis to write 'm = 3k'. Which completion states the witness guaranteed by divisibility?

    • for every real number k
    • for some positive integer k
    • for some integer k
    • for k = m in every case

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

    • There is one problem P such that every student S solved P, with P fixed for the whole group.
    • There is a student S who solved every problem P, with that student fixed for the whole task list.
    • For every student S there is a problem P that S solved.

  10. Negate the claim 'Every reader can check at least one step.' Which sentence states exactly what must happen for that claim to fail?

    • There is a reader who can check every proof step.
    • There is a reader who cannot check any step.
    • Every reader is unable to check any proof step.
    • There is a proof step that no reader can check.
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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. Introduce a,b; choose r,s; add the expressions; conclude evenness. · Introduce the even integers, obtain their integer witnesses, add the expressions, and conclude that the sum is twice an integer.
  2. We prove directly that two odd integers have an even sum. Let a and b be odd integers. · 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. The student-first claim holds; the task-first claim fails. · Each student has a solved task. Neither P nor Q was solved by both students, so there is no common solved task.
  6. For every integer x there is an integer y with y > x. · After any integer x is given, y = x + 1 works. No fixed integer y is greater than every integer x.
  7. At the deduction: since n is prime and n > 2, it is odd, because an even prime is 2. · Cite primality and n > 2 at the parity deduction. The prime 2 shows why the size condition matters.
  8. for some integer k · Divisibility by 3 supplies an integer k with m = 3k. It does not require k to be positive or equal to m.
  9. For every student S there is a problem P that S solved. · Putting the student first allows the problem to depend on that student. It does not require different students to solve different problems.
  10. There is a reader who cannot check any step. · Failure requires one reader with no checkable step. It does not require this of every reader or require a step that defeats all readers.
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