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PROOF
If you are asked to ‘prove’ (or disprove) a mathematical statement, what does this actually mean? The Oxford English Dictionary defines proof as ‘The action or an act of testing something; a test, a trial, an experiment … An operation to check the correctness of an arithmetical calculation.’ Can you show that if x and y are even integers then the sum of x and y will also be an even integer? For a group of mathematical statements to constitute a proof you need to ensure that the proof will work for all possibilities and not just one or two. For this proof, how could you ensure that x and y are even integers? Well, an even integer is a whole number that is divisible by 2. So if we say that m, n, p ... are integers, then 2m will be an even integer and so will 2n, 2p, etc. Now you can let x = 2m and y = 2n. If you now sum x and y you get x + y = 2m + 2n. This simplifies to 2(m + n), which is an even number. You have now proved that if x and y are even integers then the sum of x and y will also be an even integer. If you are asked to ‘show’, ‘prove’ or ‘demonstrate’, you are being asked to prove that something is true. The word ‘prove’ may not actually appear in what you are being asked to do but you need to understand that this is what is required of you. You may also be asked to ‘disprove’ something, or to ‘show by using a counter example’ that something is false. Throughout this book, questions involving proof are flagged as such and proof is discussed in context in the margin.
LEARNING OBJECTIVES You will learn how to:
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understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion
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use proof by deduction use proof by exhaustion use disproof by counter example.
TOPIC LINKS Most chapters of this book demonstrate proofs related to specific topics within those chapters. For example, Chapter 2 Algebra and functions 2: Equations and inequalities includes the proof for the quadratic formula by completing the square. In this chapter you will practise using different types of proof.
PRIOR KNOWLEDGE You should already know how to:
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argue mathematically to show that algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs
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