Proof (DP IB Analysis & Approaches (AA)): Revision Note

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

• The set of positive integers is denoted by

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  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

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

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

•  – the set of natural numbers

•  – the set of integers

•  – the set of quotients (rational numbers)

•  – the set of real numbers

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

  • It is the only number that satisfies 

• 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