Counterexamples and Testing a Conjecture · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Breaking claims to make them true

Mathematics · Mathematical Thinking · ages 22-24
Name ______________________   Date ____________

  1. Claim: all prime numbers are odd. Which counterexample breaks it?

    • 2
    • 9
    • 15
    • 1

  2. Claim: all prime numbers are odd. Which counterexample breaks it?

    • 9
    • 2
    • 15

  3. What is the negation of the claim all A are B?

    • Some A is not B
    • No A is B
    • All B are A

  4. Testing a conjecture at n equals 0 and n equals 1 is a waste of time; only large random cases matter.

    Circle one:   True   False

  5. Tom tries hard to break the claim that differentiability at a point forces continuity there, and every candidate function defeats him. What does this pattern of failures point to?

    • Differentiability genuinely forces continuity, so that hypothesis is doing the real work
    • Continuity forces differentiability
    • No differentiable functions exist
    • Proofs never need hypotheses

  6. Every attempt to break the claim that differentiability at a point forces continuity there collapses, always at continuity. What does this pattern point to?

    • Continuity forces differentiability
    • No differentiable functions exist
    • Differentiability genuinely forces continuity, so that hypothesis does the real work

  7. Claim: the product of any two irrational numbers is irrational. Which pair breaks it?

    • Root 2 times root 3
    • Root 5 times root 7
    • Root 2 times root 2

  8. Claim: the product of any two irrational numbers is irrational. Which pair breaks it?

    • root 2 times root 2
    • root 2 times root 3
    • pi times 2
    • root 5 times root 7

  9. Claim: if f and g are both discontinuous at a point, then f plus g is discontinuous there. Hunting keeps landing on cancelling pairs like g equal to minus f. Which repaired claim is actually true?

    • If f is continuous and g is discontinuous at a point, then f plus g is discontinuous there
    • If f is discontinuous at a point, then f plus f is discontinuous there
    • If f is continuous, then f times f is discontinuous

  10. Claim: if f and g are both discontinuous at a point, then f plus g is discontinuous there. Hunting for a counterexample keeps landing on pairs that cancel out, like g equal to minus f. Which repaired claim is actually true?

    • If f is continuous and g is discontinuous at a point, then f plus g is discontinuous there
    • If f is discontinuous at a point, then f plus f is discontinuous there
    • If f and g are discontinuous at different points, then f plus g is discontinuous everywhere
    • If f is continuous, then f times f is discontinuous
LightMySky · lightmysky.comW1-mt_SjA4jqNkaM-s1

Answer key

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

Breaking claims to make them true W1-mt_SjA4jqNkaM-s1

  1. 2 · 2 is prime and even, so it meets the setup but breaks the claimed result.
  2. 2 · 2 is prime and even, so it meets the setup but breaks the result.
  3. Some A is not B · Denying a universal claim needs exactly one object breaking the pattern.
  4. False · Small and awkward cases break most false conjectures fastest.
  5. Differentiability genuinely forces continuity, so that hypothesis is doing the real work · When every attack fails at the same hypothesis, that hypothesis is exactly what makes the theorem true.
  6. Differentiability genuinely forces continuity, so that hypothesis does the real work · When every attack fails at the same hypothesis, that hypothesis makes the theorem true.
  7. Root 2 times root 2 · Both factors are irrational, yet root 2 times root 2 equals 2.
  8. root 2 times root 2 · Both factors are irrational, yet root 2 times root 2 equals 2, which is rational.
  9. If f is continuous and g is discontinuous at a point, then f plus g is discontinuous there · If f plus g were continuous while f is continuous, g would be continuous too.
  10. If f is continuous and g is discontinuous at a point, then f plus g is discontinuous there · If f plus g were continuous while f is continuous, then g equal to (f plus g) minus f would be continuous too, a contradiction.
Worksheet · LightMySky