Deduction, contradiction and counterexamples
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| counterexample/ˈkaʊntəreɡzæmpl/ | Counterexample |
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.
The method
- 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.
Why does n=41 disprove "n²+n+41 is prime for every positive integer n"?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A universal claim falls to one counterexample. At n=41 the value is 41×43, divisible by 41, so not prime - no further cases are needed.
A worked case
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
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
At n=41 the value is 41²+41+41=41×43=1763.
Where it goes wrong
- 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.
Not true: Checking the first ten integers proves a claim for every positive integer. A pattern can hold for the first ten cases and then fail; a proof must cover every case.
For 2a+1 and 2b+1 with a=3,b=4, find their sum.
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The two odd integers are 7 and 9, whose sum is 16.
Checking the first ten integers proves a claim for every positive integer.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A pattern can hold for the first ten cases and then fail; a proof must cover every case.
Applying it
- 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.
Find 1+2+...+20.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Pair the endpoints: 20 terms have average 10.5, giving 20×10.5=210.
Match each proof method to its structure.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Each method has a required structure; checking many cases is evidence but never a proof of a universal claim.
In your course
- AQA A-level Mathematics (7357), specification section A.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Counterexample: A single valid case that disproves a universal claim.
- A deductive proof starts from stated definitions or assumptions.
- Watch for: do not assume the conclusion while proving it.