Deduction, contradiction and counterexamples · Déduction, contradiction et contre-exemples
| English | Français |
|---|---|
| counterexample/ˈkaʊntəreɡzæmpl/ | contre-exemple |
When does a pattern become a proof?
- Checking several integers can suggest a pattern. What turns the pattern into a proof for every integer?
- This lesson studies counterexample 反例: A single valid case that disproves a universal claim.
Choose the mathematical structure
- A deductive proof starts from stated definitions or assumptions. A counterexample refutes an all-values claim. For contradiction, assume the contrary and derive an impossibility. For exhaustion, check every case in a finite complete list.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines counterexample? · Laquelle des descriptions définit correctement un contre-exemple ?
A single valid case that disproves a universal claim. · Un seul cas valide qui infirme une affirmation universelle.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Odd integers 2a+1 and 2b+1 sum to 2(a+b+1), hence even. To disprove that n²+n+41 is always prime, use n=41: the value is 41×43=1763. In a contradiction proof that √2 is irrational, assume √2=p/q in lowest terms; p²=2q² forces both p and q even, contradicting lowest terms.
Deduction, contradiction and counterexamples · Déduction, contradiction et contre-exemples
A deductive proof starts from stated definitions or assumptions · Une preuve déductive part de définitions ou d'hypothèses données
Distinguish a proof for all integers from one counterexample. · Distinguer une preuve valable pour tous les entiers d'un seul contre-exemple.
Evaluate n²+n+41 at n=41. · Évaluez n²+n+41 pour n=41.
At n=41 the value is 41²+41+41=41×43=1763. · Pour n=41, la valeur est 41²+41+41=41×43=1763.
Test a tempting shortcut
- Do not assume the conclusion while proving it. In contradiction, identify the impossible consequence and reject the initial contrary assumption. Induction is not compulsory in AQA 7357; it belongs in its own appropriate further-mathematics scope.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Checking the first ten integers proves a claim for every positive integer. This claim is false. Explain which definition or assumption it violates.
For 2a+1 and 2b+1 with a=3,b=4, find their sum. · Pour 2a+1 et 2b+1 avec a=3, b=4, trouvez leur somme.
The two odd integers are 7 and 9, whose sum is 16. · Les deux entiers impairs sont 7 et 9, dont la somme est 16.
Checking the first ten integers proves a claim for every positive integer. · Vérifier les dix premiers entiers prouve une affirmation pour tout entier positif.
Do not assume the conclusion while proving it. In contradiction, identify the impossible consequence and reject the initial contrary assumption. Induction is not compulsory in AQA 7357; it belongs in its own appropriate further-mathematics scope. · Ne pas supposer la conclusion lors de sa démonstration. En contradiction, identifier la conséquence impossible et rejeter l'hypothèse initiale contraire. L'induction n'est pas obligatoire dans A-Level 7357 ; elle relève d'un programme propre de mathématiques supplémentaires appropriées.
Interpret a new situation
- AQA proof includes deduction, exhaustion and contradiction. IAL P2 introduces exhaustion and counterexample, while P4 introduces contradiction. Keep the method matched to the named unit.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find 1+2+...+20. · Trouver 1+2+...+20.
Pair the endpoints: 20 terms have average 10.5, giving 20×10.5=210. · Appairer les extrémités : 20 termes ont une moyenne de 10.5, donnant 20×10.5=210.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
Use this in your course
- 7357 · A-level · A. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A single valid case that disproves a universal claim. Choose the relationship, show the method, check its assumptions and interpret the result.