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

By Peter J. Eccles

This e-book eases scholars into the pains of collage arithmetic. The emphasis is on realizing and developing proofs and writing transparent arithmetic. the writer achieves this via exploring set conception, combinatorics, and quantity thought, issues that come with many basic rules and should no longer join a tender mathematician's toolkit. This fabric illustrates how standard principles might be formulated conscientiously, offers examples demonstrating quite a lot of easy tools of evidence, and comprises a number of the all-time-great vintage proofs. The publication offers arithmetic as a regularly 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 scholar but additionally lots of regimen routines to assist familiarize the reader with the elemental rules.

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Extra resources for An Introduction to Mathematical Reasoning : Numbers, Sets and Functions

Example text

It includes a leisurely discussion of universal and existential statements which are so important in university mathematics, particularly analysis or advanced calculus. This is not a book on mathematical logic but inevitably some ideas from the beginnings of that subject are included in these first two parts. The reader may find some of the material on counting in Part III to be more difficult than the rest of the book. It is best not to become discouraged by this but if necessary to move on to Part IV which is probably the easiest part but also includes some very striking and attractive results, returning to Part III when more familiarity with the language of sets and functions has been acquired.

I) P is equivalent to Q. (ii) P is necessary and sufficient for Q. (iii) P if and only if Q (sometimes written P iff Q). (iv) P precisely when Q. 2 Arithmetic The basic principles of mathematical reasoning apply throughout mathematics but in this book we are going to explore them mainly in the context of number theory or arithmetic since the basic ideas here are familiar from early schooldays. e, the numbers … , –2, –1, 0, 1, 2, 3, … , and their basic arithmetic properties (under addition and multiplication) and order properties.

The justification for such steps usually makes use of the idea of ‘implication’; an implication is the assertion that if one particular statement is true then another particular statement is true. 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.

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