**How To Master Techniques Of Proof For AS Maths **

Proof questions at AS Level require you to apply **simple mathematical rules** to show that certain statements are universally true (or untrue).

They are almost like **logic puzzles** for mathematicians, so they can be quite fun (honestly)!

Once you’ve become familiar with the ways that examiners **phrase** the questions and you’ve learnt the right **notation** to use, proof questions can be a piece of cake.

Read on to find everything you need to know about AS Proof questions, straight from the brain of a Save My Exams Maths teacher. Remember, if you’re looking to get ahead of the class, don’t skip over the **worked examples**!

*This blog post has been adapted from our new teacher-written **OCR AS Maths: Pure Revision Notes collection**.*

**Notation in Proof **

The below diagram provides a helpful guide to the notation which you should use when ‘proving’. Don’t try to invent your own – the examiners won’t give you any marks!

Remember – the term *integer* refers to a whole number. A *set of integers* is a group of whole numbers.

**Proof by Exhaustion **

This method literally requires you to ‘exhaust’ the possibilities – i.e. check all values – when proving something to be true. It’s important to only use this technique when it’s sensible to do so – don’t be testing tens (or hundreds) or values!

An example:

*The set of numbers S is defined as all positive integers greater than 5 and less than 10. Prove by exhaustion that the square of all values in S differ from a multiple of 5 by 1. *

**Proof by Counter Example **

To prove that a statement is **untrue**, you just need to find **one exception** to the rule.

You might also be asked to **disprove** a statement – but the same method applies.

You’ll need to get your detective hat on to find the anomaly! The below diagram provides some useful tips for finding this value…

An example:

*Use a counter example to prove that the difference between any two square numbers is not always odd*

(an EVEN number, proving the statement)

Right, that’s the theory covered. Let’s go deeper into this topic with three **worked examples**. Give them a try for yourself** before **looking at the answers!

**Question 1**

*Prove that the sum of any three consecutive even numbers is always a multiple of 2, but not always a multiple of 4. *

Three even consecutive numbers =

Their sum =

Factorised = – Which is **always a multiple of 2, but not always a multiple of 4**!

**Question 2**

*Prove that 23 is a prime number*

There is a rule that all numbers can be written as a product of their prime factors. So, all non-primes should have at least one other factor (besides themselves and 1).

Because 23 **does not have any prime factors below** , we have proven it is a prime number.

**Question 3**

*Prove that, if is odd, ** is odd, and that if is even, ** is even *

First, factorise the equation to get

If is odd, and are also odd.

**The product of three odd numbers is always odd**

If is even, and are also even.

**The product of three even numbers is always even **

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**Are you ready to try some more practice questions? Head on over to our OCR AS Maths: Pure Topic Questions! Don’t worry if you’re not feeling 100% confident yet; every question is accompanied by a full answer explanation. **

**If you’d like to revise another topic, check out the full Revision Notes collection here. **