Disproving with a counterexample
Practice Flashcards
Flip to reveal answersWhat is a counterexample?
Track your progress — Sign up free to save your progress and get smart review reminders based on spaced repetition.
All 8 Flashcards — Disproving with a counterexample
Sign up free to track progress and get spaced-repetition review schedules.
Question
What is a counterexample?
Answer
A single case where a 'for all' statement fails — enough to prove the statement false.
Question
How many counterexamples disprove a universal statement?
Answer
Just one.
Question
Where should you look for counterexamples?
Answer
Small numbers (0, 1), negatives, fractions, and edge cases.
Question
Counterexample to 'all primes are odd'?
Answer
2 — it's prime and even.
Question
Counterexample to 'a² = b² ⇒ a = b'?
Answer
a = 2, b = −2: 4 = 4 but 2 ≠ −2.
Question
Counterexample to 'x² ≥ x for all real x'?
Answer
x = ½: ¼ < ½.
Question
Counterexample to 'the sum of two irrationals is irrational'?
Answer
√2 + (−√2) = 0, which is rational.
Question
What must you check about a counterexample?
Answer
That it meets the statement's condition (hypothesis) but breaks the conclusion.
Read the notes
Full study notes for Disproving with a counterexample
Topic 1.15 hub
Proof by induction (HL only)
More from Topic 1.15
All flashcards in this topic
Math AA exam skills
Paper structures & tips
Track your progress with spaced repetition
Sign up free — Aimnova tells you exactly which cards to review and when, so you remember everything before your IB exam.
Start Free