Deduction, contradiction and induction
| English | 中文 | Pinyin |
|---|---|---|
| counterexample/ˈkaʊntəreɡzæmpl/ | 反例 | fǎn lì |
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. Induction requires a base case and an implication from n=k to n=k+1; checking successive cases alone is not induction.
- 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?
A single valid case that disproves a universal claim.
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.
For odd integers 2a+1 and 2b+1, their sum is 2(a+b+1), hence even. The claim n²+n+41 is always prime fails at n=41: the value is 41×43=1763. For 1+...+n=n(n+1)/2, the induction step adds k+1 to the assumed sum.
Deduction, contradiction and induction
A deductive proof starts from stated definitions or assumptions
Distinguish a proof for all integers from one counterexample.
Evaluate n²+n+41 at n=41.
At n=41 the value is 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.
The two odd integers are 7 and 9, whose sum is 16.
Checking the first ten integers proves a claim for every positive integer.
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.
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.
Pair the endpoints: 20 terms have average 10.5, giving 20×10.5=210.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL mathematics; official unit P2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- 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.