Deduction, contradiction and induction
مُقدَّم| English | المعنى |
|---|---|
| counterexample/ˈkaʊntəreɡzæmpl/ | مثال مضاد |
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.
- Induction requires a base case and an implication from n=k to n=k+1; checking successive cases alone is not induction.
Why does n=41 disprove "n²+n+41 is prime for every positive integer n"?
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
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
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.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
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.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
The two odd integers are 7 and 9, whose sum is 16.
Checking the first ten integers proves a claim for every positive integer.
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
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.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Pair the endpoints: 20 terms have average 10.5, giving 20×10.5=210.
Match each proof method to its structure.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Each method has a required structure; checking many cases is evidence but never a proof of a universal claim.
In your course
- Pearson Edexcel International A Level Pure Mathematics, unit FP1.
- 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.
Write from the value 1681
- Evaluate n squared plus n plus 41 at n equals 40.
- Say what one composite value does to a claim about every n.
Evaluate n squared plus n plus 41 at n equals 40. What does that value show about the claim that the expression is prime for every positive integer n?
1600 plus 81 is 1681, a square.
What is the value at n equals 40?
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
The value is a square, so it is not prime. Other options — 1763 (the lesson value at n equals 41); 41 (the constant term).