Proof by Contradiction | A-level Maths | OCR, AQA, Edexcel скачать в хорошем качестве

Proof by Contradiction | A-level Maths | OCR, AQA, Edexcel

Proof by Exhaustion in a Snap! Unlock the full A-level Maths course at http://bit.ly/2ma7ntC created by Lewis Croney, Maths expert at SnapRevise. SnapRevise is the UK’s leading A-level and GCSE revision & exam preparation resource offering comprehensive video courses created by A* tutors. Our courses are designed around the OCR, AQA, SNAB, Edexcel B, WJEC, CIE and IAL exam boards, concisely covering all the important concepts required by each specification. In addition to all the content videos, our courses include hundreds of exam question videos, where we show you how to tackle questions and walk you through step by step how to score full marks. Sign up today and together, let’s make A-level Maths a walk in the park! The key points covered in this video include: 1. Opposite Statements 2. Structure of a Proof by Contradiction 3. Proving √2 is Irrational by Contradiction 4. Examples Opposite Statements We have been examples of proof by deduction. We are going to use a new method of proof, which involves deducing a logical inconsistency. Given a conjecture, we consider the opposite statement. If we can deduce that this statement is false, we will have proven our original conjecture. We would like to use this method to prove results that we are not able to directly. Structure of a Proof by Contradiction The process of taking the opposite of a conjecture and showing that it is not possible is called proof by contradiction. We are looking to deduce a logical inconsistency with the opposite statement. We will then have proven our original conjecture. In this case we write down our finite list of primes and form a special number from them. This special number must be divisible by a prime number, so we can consider a general prime number dividing it. We can check if this primed number is any of the primes on our original list. Hence, we have found a prime not on our original list, giving us a contradiction. This means our original conjecture must be true, and so we have proven it. Proof √2 is Irrational by Contradiction There is another famous proof by contradiction. We actually require a lemma for this proof - one which itself needs a short proof by contradiction. To prove the lemma, we consider the opposite statement. We can write the odd condition algebraically. We can then square this expression. This gives us our contradiction. So the original conjecture is true. We can now begin the main proof by assuming the opposite statement is true. We can write this opposite statement algebraically. We impose particular conditions on the values of a and b. Performing the square of both sides removes the root. Removing fractions is useful since it’s easier to work with integers. We can deduce a result about a. This is where our earlier lemma can be applied. Then we can also write this condition algebraically. We expand the square, cancel the factors, deduce a similar result with b and apply our earlier lemma again. This gives us our contradiction. Hence, we can finally deduce that our initial conjecture was correct. Summary Sometimes we may be unable to prove conjectures, even when they are true It can be helpful to consider opposites of given statements If we can show opposite statements to be logically inconsistent, then we have proven our original conjecture We often call logical inconsistencies contradictions There are famous proofs by contradictions including, existence of infinitely many primes and irrationality of √2