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IBMaths: AA SLDPRevision Notes1. Number & Algebra1.4 Proof & Reasoning1.4.1 Proof

Proof (DP IB Maths: AA SL)

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Language of Proof

 What is 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 straight integer numbers
    • The set of positive integers is denoted by straight integer numbers to the power of plus

How do we prove a statement is true for all values?

  • 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 identical to 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

Examiner Tip

  • 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 

Worked example

Prove that open parentheses 2 x minus 2 close parentheses open parentheses x minus 3 close parentheses plus 2 left parenthesis x minus 1 right parenthesis equals 2 left parenthesis x minus 2 right parenthesis left parenthesis x minus 1 right parenthesis.

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

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Proof by Deduction

What is proof by deduction?

  • A mathematical and logical argument that shows that a result is true

How do we do proof by deduction?

  • 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 left parenthesis straight natural numbers comma space straight integer numbers comma space straight rational numbers comma space straight real numbers right parenthesis
    • straight natural numbers – the set of natural numbers
    • straight integer numbers – the set of integers
    • straight rational numbers – the set of quotients (rational numbers)
    • straight real numbers – the set of real numbers

1.1.2 Proof by Deduction Notes Diagram 2

Examiner Tip

  • 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

Worked example

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

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

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.