Suppose it is false, then watch it break: Proof by contradiction has three moves: assume the statement is false, reason until you reach something impossible, then conclude the statement must be true after all.
Press play to watch the assumption break line by line.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Begin with the opposite: Start from the negation, and write “assume” or “suppose”, never “let”. The proof must open by supposing the opposite is true.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Why the primes never run out: A second flavour: instead of a parity clash, build a new object the assumption says cannot exist. This is Euclid's proof that the primes never run out.
Press play — the built number N is the object that breaks the assumption.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Build the impossible object: When a parity or factor clash is hard to see, try constructing a number or object the assumption forbids.
Memorize terms 3x faster
Smart flashcards show you cards right before you forget them. Perfect for definitions and key concepts.
There is no smallest positive fraction: A third flavour: the contradiction is finding something smaller than the supposed smallest. Short and sharp — a contradiction need not be about parity.
Press play — halving r is all it takes to break the claim.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Beat the extreme: To contradict a “smallest” or “largest” claim, produce one that is smaller (or larger) but still fits.
An exam-style question
Prove by contradiction that the equation has no integer solutions.
Step by step
- Assume the opposite. Suppose some integer is a solution.
- Move the 1 across. Take the to the other side.
- Factor the left side. . If is even this is even; if is odd then is even, so it is even again — the left side is always even.
- Spot the clash. The left side is even, but is odd. An even number can never equal an odd one.
- Conclude. The assumption leads to an impossibility, so has no integer solutions.
Final answer
No integer solutions (an even number cannot equal the odd ).
To get full marks
- Start by assuming an integer solution exists, so .
- Show the left side is always even, while is odd.
- State the contradiction and conclude there is no integer solution.