37: Proof37: Proof

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”Vol. 2: A2 Core Vol. 2: A2 Core

ModulesModules

Proof

AQAModule C3

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Module C4AQA

MEI/OCROCR

Edexcel

OCR

ProofProof 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.

ProofProof by ContradictionMethod: Assume the result is NOT true and show that this leads to a contradiction.

xx 212 e.g. Prove that, if x is real, Proof:

0212 xxSubtract 2x:

Factorise:

xx 212 So,Tip: Inequalities are usually easier to deal

with if there is zero on the r.h.s.0122 xx

The squared term suggests the next step. 0)1)(1( xx

Suppose is less than

12 x x2

Proof

0)1( 2 x

However, the square of real numbers is always

0so we have a contradiction.

The l.h.s. is a perfect square:

We have therefore proved thatfor all real x.

xx 212

0)1)(1( xx

ProofProof by ExhaustionMethod: We show that every case of what we want to prove must be true.e.g. Prove

that 23333 )...321(...321 nn

for the positive integers from 1 to 5 inclusive.Proof

: :1n 113

l.h.s. r.h.s. 112

:2n 921 33 93)21( 22

:3n 36321 333 366)321( 22

ProofProof

: :1n 113

l.h.s. r.h.s. 112

:2n 921 33 93)21( 22

:3n 36321 333 366)321( 22

:4n1004321 3333 l.h.s.

r.h.s. 10010)4321( 22 :5n

22554321 33333 l.h.s. r.h.s. 22515)54321( 22

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.

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:

22 baba is not true.Any pair of negative numbers with a > b will do so our counter-example could be

2,1 ba ( so a > b )Then, 12 a 42 band 22 ba so,

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.

ProofProof 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.

ProofProof by ContradictionMethod: Assume the result is NOT true and show that this leads to a contradiction.

xx 212 e.g. Prove that, if x is real, Proof:

0212 xxSubtract 2x:

Factorise:

xx 212 So,Tip: Inequalities are usually easier to deal

with if there is zero on the r.h.s.

0122 xxThe squared term suggests the next step. 0)1)(1( xx

Suppose is less than

12 x x2

Proof

0)1( 2 x

However, the square of real numbers is always

0so we have a contradiction.

The l.h.s. is a perfect square:

We have therefore proved thatfor all real x.

xx 212

0)1)(1( xx

ProofProof by ExhaustionMethod: We show that every case of what we want to prove must be true.e.g. Prove

that 23333 )...321(...321 nn

for the positive integers from 1 to 5 inclusive.Proof

: :1n 113

l.h.s. r.h.s. 112

:2n 921 33 93)21( 22

:3n 36321 333 366)321( 22

ProofProof: :1n 113

l.h.s. r.h.s. 112

:2n 921 33 93)21( 22

:3n 36321 333 366)321( 22 :4n

1004321 3333 l.h.s. r.h.s. 10010)4321( 22 :5n

22554321 33333 l.h.s. r.h.s. 22515)54321( 22

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.

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:

22 baba is not true.Any pair of negative numbers with a > b will do so our counter-example could be

2,1 ba ( so a > b )Then, 12 a 42 band 22 ba so,