An Introduction to Mathematical Reasoning: Numbers, Sets and by Peter J. Eccles

By Peter J. Eccles

This e-book eases scholars into the trials of collage arithmetic. The emphasis is on knowing and developing proofs and writing transparent arithmetic. the writer achieves this by way of exploring set conception, combinatorics, and quantity thought, issues that come with many basic rules and will now not join a tender mathematician's toolkit. This fabric illustrates how widely used principles could be formulated conscientiously, presents examples demonstrating quite a lot of simple equipment of evidence, and contains a few of the all-time-great vintage proofs. The booklet offers arithmetic as a always constructing topic. fabric assembly the desires of readers from quite a lot of backgrounds is incorporated. The over 250 difficulties contain inquiries to curiosity and problem the main capable pupil but in addition lots of regimen workouts to assist familiarize the reader with the fundamental rules.

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1 states that a < b is a sufficient condition for 4ab < (a + b)2. Is this condition also necessary? If so, prove it. If not, find a necessary and sufficient condition. 8 Prove that for all real numbers a and b, † In David Tall (editor), Advanced mathematical thinking, Kluwer, 1991. † The idea of using these ‘given–goal diagrams’ comes from the book by Daniel J. Velleman, How to prove it, a structured approach, Cambridge University Press, 1994. It extends usefully the approach taught to the author at school of starting by stating clearly what we are ‘required to prove’ or ‘required to find’.

The symbol usually used to denote implication in pure mathematics‡ is although there are a variety of forms of words which convey the same meaning. For the moment we can think of ‘P Q’ as asserting that if statement P is true then SQ is statement Q, which is often read as ‘P implies Q’. The meaning will be made precise by means of a truth table. Before doing this it is necessary to clarify what this meaning should be and to do this we consider an example concerning an integer n. Suppose that P(n) is the statement ‘n > 3’ and Q(n) is the statement ’n > 0’, where n is an integer.

1 Mathematical statements It is quite difficult to give a precise formulation of what a mathematical statement is and this will not be attempted in this book. The aim here is to enable the reader to recognize simple mathematical statements. First of all let us consider the idea of a proposition. A good working criterion is that a proposition is a sentence which is either true or false (but not both). For the moment we are not so concerned about whether or not propositions are in fact true. Consider the following list.

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