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IB DP Maths: AA HL復習筆記1.4.1 Proof

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月11日下午2:48

Proof is a series of logical steps which show a result is?true?for all specified numbers
- ‘Seeing’ that a result works for a few numbers is not enough to?show?that it will work for all numbers
- Proof allows us to show (usually algebraically) that the result will work for?all values
You must be familiar with the notation and language of proof
LHS and RHS are standard abbreviations for left-hand side and right-hand side
Integers?are used frequently in the language of proof

Most of the time you will need to use algebra to show that the left-hand side (LHS) is the same as the right-hand side (RHS)
- You?must not?move terms?from one side to the other
- Start with one side (usually the LHS) and manipulate it to show that it is the same as the other
A?mathematical identity?is a statement that is true for all values of?x?(or?θ?in trigonometry)
- If you see this symbol then you can use proof methods to show it is true
You can complete your proof by stating that RHS = LHS or writing QED

You will need to show each step of your proof clearly and set out your method in a logical manner in the exam
- Be careful not to skip steps

1-4-1-aa-sl-language-of-proof-we-solution

A proof by deduction question will often involve showing that a result is true for all integers, consecutive integers or even or odd numbers
- You can begin by letting an integer be?n
  - Use conventions for even (2n?) and odd (2n – 1) numbers

1.1.2-Proof-by-Deduction-Notes-Diagram-2

Try the result you are proving with a few different values
- Use a sequence of them (eg 1, 2, 3)
- Try different types of numbers (positive, negative, zero)
This may help you see a pattern and spot what is going on

Prove that the sum of any two consecutive odd numbers is always even.

1-4-1-aa-sl-proof-by-deduction-we-solution

Disproving a result involves finding a value that does?not?work in the result
That value is called a?counter-example

You only need to find?one?value that does not work
Look out for the set of numbers for which the statement is made, it will often be just integers or natural numbers
Numbers that have unusual results are often involved
- It is often a good idea to try the values 0 and 1 first as they often behave in different ways to other numbers
- The number 2 also behaves differently to other even numbers
  - It is the only even prime number
- If it is the set of real numbers consider how rational and irrational numbers behave differently
- Think about how positive and negative numbers behave differently
  - Particularly when working with inequalities