# AS Maths Proof

This AS Maths course (based originally on the Edexcel specification) teaches the fundamentals of the Proof topics taught during the first year of A-Level Maths. These topics include:

- Proof by Deduction
- Proof by Exhaustion
- Disproof by Counterexample.

There are 17 lessons which include notes, examples and videos for the above topics followed by an assignment section with a quiz and a written assignment at the end of the course.

We recommend that you allow 3 hours for completion of the AS Maths Proof course: approximately 1 hour for each of the sections (although some sections are longer than others). This doesn’t include the time you should allow yourself to complete the assignment – this will take around 90 minutes but that is up to you!

Mark the bottom of each lesson ‘complete’ when you feel happy that you have learned the content. Happy Studying!

### Introduction

### Proof by Deduction

Read these notes on Proof by Deduction before checking out the examples and videos or attempting the quiz and assignment.

Two written examples of Proof by Deduction with show/hide options in order to try them out for yourself first.

How to use Proof by Deduction to show that the given expression is always even, regardless of the choice of integer to put in.

Sample Assessment question requiring the use of Proof by Deduction to show that an algebraic inequality holds followed by a Disproof by Counterexample question.

Proof of the Quadratic Formula using Proof by Deduction

By recognising a hidden quadratic, Proof by Deduction can be used to show tricky inequalities.

### Proof by Exhaustion

Read these notes on Proof by Exhaustion before checking out the examples and videos or attempting the quiz and assignment.

Two written examples of Proof by Exhaustion with show/hide options in order to try them out for yourself first.

A detailed Proof by Exhaustion example showing that all square numbers are either a multiple of 4 or one more than a multiple of 4.

Another example of Proof by Exhaustion, slightly more involved than the previous video, showing that all square numbers are either a multiple of 3 or one more than a multiple of 3.

### Disproof by Counterexample

Read these notes on Disproof by Counterexample before checking out the examples and videos or attempting the quiz and assignment.

Three written examples of Disproof by Counterexample with show/hide options in order to try them out for yourself first.

This video shows the solutions that a student should give when tackling a basic disproof by counterexample question.

This video explores a proof exam question where the first part is a standard Proof by Deduction question and the second part requires the student to decide which proof to use - it turns out the statement isn't true and a simple counterexample will suffice to disprove the statement.

### Assignments

This assignment contains 16 statements in 12 questions. Either prove (by deduction or exhaustion) or disprove (by counterexample) each of the statements in the spaces provided. It is up to you to decide which proof/disproof to use.

If you would like to sit this assignment like an exam, give yourself 90 minutes to complete the tasks.

Full written solutions to the AS Maths Proof Assignment

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