Proofs
Maths: Edexcel IGCSE Higher, topic 2.1.7
Proofs
To prove an 'identity', show one side is the same as the other. The identity symbol is ≡. E.g - Show that (n + 3)(n + 2) - 3n + 2 ≡ n2 + 2n + 8:
Show these are the same
Begin with the left hand side (LHS)
Expand brackets
Collect terms
This is the same as the right hand side so we are done
Proofs and Counter Examples
General numbers in proofs
- If we are proving something involving an even number, use 2n.
- If we are proving something involving an odd number, use 2n + 1.
- Consecutive numbers are shown by n, n + 1, n + 2, ... etc
Counter examples
- A counter example is the simplest way to prove a statement is wrong.
- You only need one counter example to show something is wrong.
- To disprove the statement ‘The product of two primes is always odd':
- Use the counter example of the primes 2 and 3 whose product is 6 which is even.
Test yourself on Proofs
What is the general expression for even numbers?
What is the general expression for odd numbers?
What is the general expression for consecutive numbers?
Give a counter example to disprove the statement:
x² is always more than xGive a counter example to disprove the statement:
Any number larger than 11 is divisible by 3