Contradiction, Counterexample and Proof Chains
Use contradiction and counterexample, develop conjectures, order proof statements and sustain sophisticated reasoning chains.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Assume the negation of a target and derive an impossibility for proof by contradiction.
Exam cue: For contradiction, state exactly which target is being negated.
Concept 2
Use small cases to form a conjecture, then replace observation with justification.
Exam cue: For disproof, seek one permitted counterexample.
Concept 3
Order or construct a proof by tracking dependencies between statements.
Exam cue: When ordering steps, identify which facts each statement needs.
Risk pitfalls and guardrails
Treating several successful examples as a proof.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Producing an example that falls outside the stated domain.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Reordering algebraic steps without checking reversibility.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
Contradiction start
Assume the precise negation of the desired conclusion.
Contradiction finish
Derive an impossibility, so the negated assumption fails.
Counterexample
One in-domain failure disproves a universal claim.
Conjecture
Small cases suggest a claim but do not justify it.
Dependency order
A proof step must appear after every fact it uses.
Long chain
Preserve conditions and implication directions at every link.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Arrange a proof that the square of an even integer is divisible by 4. What comes first?
After 'let n=2k', which step follows?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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