1. 37: Proof © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

2. Proof AQA Module C3 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C4 AQA MEI/OCR OCR Edexcel OCR

3. Proof Proof is a fundamental part of mathematics. By now you understand the difference between proving a result and showing that it appears to be true. The proofs you have met are examples of direct proof and you should be able to carry out simple ones yourself. These are likely to arise in the trig identities. Other methods of proof ( and one of disproof ) are referred to in the specifications and I am going to give you one example of each method. The methods may not be examined. If you are in any doubt it would be advisable to contact your exam board for clarification.

4. Proof Proof by Contradiction Method: Assume the result is NOT true and show that this leads to a contradiction. e.g. Prove that, if x is real, Proof: Subtract 2x : Factorise: So, Tip: Inequalities are usually easier to deal with if there is zero on the r.h.s. The squared term suggests the next step. Suppose is less than x2

5. Proof However, the square of real numbers is always so we have a contradiction. The l.h.s. is a perfect square: We have therefore proved that for all real x.

6. Proof Proof by Exhaustion Method: We show that every case of what we want to prove must be true. e.g. Prove that for the positive integers from 1 to 5 inclusive. Proof: l.h.s.r.h.s.

7. Proof Proof: l.h.s.r.h.s. l.h.s. r.h.s. l.h.s. r.h.s.

8. Proof I’ve proved the result for the integers from 1 to 5 and it looks as though it will always be true BUT I have NOT proved it is always true. I would be exhausted even proving it by this method for integers up to 10. Proving it for all positive integers by the method of exhaustion is impossible.

9. Disproof This is a method for proving a statement is NOT true. Disproof by Counter-Example Method: We find ONE example where the statement does not hold and we have done enough to show that it is not always true. e.g. Show that the statement: is not true. Any pair of negative numbers with a > b will do so our counter-example could be ( so a > b ) Then,andso,

10. Proof The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

11. Proof Proof is a fundamental part of mathematics. By now you understand the difference between proving a result and showing that it appears to be true. The proofs you have met are examples of direct proof and you should be able to carry out simple ones yourself. These are likely to arise in the trig identities. Other methods of proof ( and one of disproof ) are referred to in the specifications and I am going to give you one example of each method. The methods may not be examined. If you are in any doubt it would be advisable to contact your exam board for clarification.

12. Proof Proof by Contradiction Method: Assume the result is NOT true and show that this leads to a contradiction. e.g. Prove that, if x is real, Proof: Subtract 2x : Factorise: So, Tip: Inequalities are usually easier to deal with if there is zero on the r.h.s. The squared term suggests the next step. Suppose is less than x2

13. Proof However, the square of real numbers is always so we have a contradiction. The l.h.s. is a perfect square: We have therefore proved that for all real x.

14. Proof Proof by Exhaustion Method: We show that every case of what we want to prove must be true. e.g. Prove that for the positive integers from 1 to 5 inclusive. Proof: l.h.s.r.h.s.

15. Proof Proof: l.h.s.r.h.s. l.h.s. r.h.s. l.h.s. r.h.s.

16. Proof I’ve proved the result for the integers from 1 to 5 and it looks as though it will always be true BUT I have NOT proved it is always true. I would be exhausted even proving it by this method for integers up to 10. Proving it for all positive integers by the method of exhaustion is impossible.

17. Disproof This is a method for proving a statement is NOT true. Disproof by Counter-Example Method: We find ONE example where the statement does not hold and we have done enough to show that it is not always true. e.g. Show that the statement: is not true. Any pair of negative numbers with a > b will do so our counter-example could be ( so a > b ) Then,andso,