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Flip to reveal answersHow does proof by contradiction work?
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All 8 Flashcards — Proof by contradiction
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Question
How does proof by contradiction work?
Answer
Assume the statement is false, derive something impossible (a contradiction), so the assumption is wrong and the statement is true.
Question
What do you assume at the start of a contradiction proof?
Answer
The negation (opposite) of what you want to prove.
Question
What does reaching a contradiction prove?
Answer
That the assumption (the opposite) is false — so the original statement is true.
Question
Outline the proof that √2 is irrational.
Answer
Assume √2 = p/q in lowest terms; show p² = 2q² makes both p and q even, contradicting 'no common factor'.
Question
How do you prove 'if n² is even then n is even' by contradiction?
Answer
Assume n is odd (n = 2k+1); then n² = 2(2k²+2k)+1 is odd, contradicting n² even.
Question
Prove the sum of a rational and an irrational is irrational — the contradiction?
Answer
Assuming r + x is rational forces x = (r + x) − r to be rational, contradicting x irrational.
Question
How should you open a contradiction proof in an exam?
Answer
'Assume, for contradiction, that … [the negation].'
Question
Is a contradiction a mistake?
Answer
No — it's the goal; it shows the assumption can't hold.
Read the notes
Full study notes for Proof by contradiction
Topic 1.15 hub
Proof by induction (HL only)
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Math AA exam skills
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