1.4.1 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
  • The set of positive integers is denoted by

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  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

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
  •  – the set of natural numbers
  •  – the set of integers
  •  – the set of quotients (rational numbers)
  •  – the set of real numbers

What is proof by exhaustion?

• Proof by exhaustion is a way to show that the desired result works for every allowed value
  • This is a good method when there are only a limited number of cases to check
• Using proof by exhaustion means testing every allowed value not just showing a few examples
  • The allowed values could be specific values
  • They could also be split into cases such as even and odd

What is disproof by counter-example?

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

How do I disprove a result?

• 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