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IB DP Maths: AA HL復習筆記 1.5.2 Proof by Contradiction

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月11日下午2:59

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

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

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
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

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
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

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