IB DP Maths: AA HL复习笔记 1.5.2 Proof by Contradiction
What is proof by contradiction?
- Proof by contradiction?is a way of proving a?result is true?by showing that?the negation can not be true
- It is done by:
- Assuming the negation (opposite) of the result is true
- Showing that this then leads to a contradiction
How do I determine the negation of a statement?
- The?negation?of a statement is the?opposite
- It is the statement that makes the original statement false
- To negate statements that mention “all”, “every”, “and” “both”:
- Replace these phrases with “there is at least one”, “or” or “there exists” and include the opposite
- To negate statements that mention “there is at least one”, “or” or “there exists”:
- Replace these phrases with “all”, “every”, “and” or “both” and include the opposite
- To negate a statement with “if A occurs then B occurs”:
- Replace with “A occurs and the negation of B occurs”
- Examples include:
| Statement | Negation |
| a?is?rational | a?is?irrational |
| every?even number bigger than 2?can be written?as the sum of two primes | there exists?an even number bigger than 2 which?cannot be written?as a sum of two primes |
| n?is?even?and?prime | n?is?not even?or?n?is?not prime |
| there is at least one?odd?perfect number | all?perfect numbers are?even |
| n?is a multiple of 5?or?a?multiple of 3 | n?is not a multiple of 5?and?n?is not a multiple of 3 |
| if?n2?is even?then?n?is even | n2?is even?and?n?is odd |
What are the steps for proof by contradiction?
- STEP 1: Assume the negation?of the statement is?true
- You assume it is true but then try to prove your assumption is wrong
- For example: To prove that there is no smallest positive number you start by assuming there is a smallest positive number called?a
- You assume it is true but then try to prove your assumption is wrong
- STEP 2:?Find?two results which contradict?each other
- Use algebra to help with this
- Consider how a contradiction might arise
- For example: ?a?is positive and it is smaller than?a?which contradicts that?a?was the smallest positive number
- STEP 3:?Explain?why the?original statement is true
- In your explanation mention:
- The?negation can’t be true?as it led to a contradiction
- Therefore the?original statement must be true
- In your explanation mention:
What type of statements might I be asked to prove by contradiction?
-
Odds and evens
- To show that?n?is even if?n2?is even
- Assume?n2?is even and?n?is odd
- Show that?n2?is odd
- To show that?n?is even if?n2?is even
- Maximum and minimum values
- To show that there is no maximum multiple of 3
- Assume there is a maximum multiple of 3 called?a
- Multiply?a?by 3
- To show that there is no maximum multiple of 3
Exam Tip
- A question won't always state that you should use proof by contradiction, you will need to recognise that it is the correct method to use
- There will only be two options (e.g. a number is rational or irrational)
- Contradiction is often used when no other proof seems reasonable
Worked Example
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