Proof | DP IB Maths: AA SL Revision Notes 2021

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Language of Proof

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
- The set of integers is denoted by $ℤ$
- The set of positive integers is denoted by $ℤ^{+}$

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)
- The symbol $\equiv$ is used to identify an identity
- 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

Prove that $(2 x - 2) (x - 3) + 2 (x - 1) = 2 (x - 2) (x - 1)$ .

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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
You will need to be familiar with sets of numbers $(ℕ, ℤ, ℚ, ℝ)$
- $ℕ$ – the set of natural numbers
- $ℤ$ – the set of integers
- $ℚ$ – the set of quotients (rational numbers)
- $ℝ$ – the set of real 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.

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