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Revision guide Contents Introduction Pointless module The pedantic mathematician Disclaimer Acknowledgements Logic Truth tables Universe of Discourse Numbers numbers all kinds of numbers The naturals The integers The rationals The irrationals The reals Th

Contents Introduction Pointless module The pedantic mathematician Disclaimer Acknowledgements Logic Truth tables Universe of Discourse Numbers numbers all kinds of numbers The naturals The integers The rationals The irrationals The reals The complex numbers C Domino-like proof and the FUNdaMENTAL Theorem of Arithmetic Primes The Well-Ordering Principle Proving stu ? with the WOP Proof by Induction A slight variant of the Principle of Induction The FUNdaMENTAL Theorem of Arithmetic HCF and LCM Euclidean Algorithm The extended Euclidean Algorithm Integers and Modular Arithmetic Subgroups of Z The wonders of Modular Arithmetic Congruence A last thing you should know Some Fun Stu ? Multiplicative Inverse Gaussian Algorithm Rationality Irrationality Reality Decimal Expansions ??to in ?nity and beyond ? - Buzz Lightyear Going Beyond Reality The Complex Realm The Complex Plane The Conjugate The Argument and Absolute Value Evaluating the Argument and the Absolute Value Some Properties of z Polar Co-ordinates The Exponential Form Roots of numbers Using De Moivre ? s Theorem Using the Exponential Form of Complex Numbers nth Roots of Unity Functions Injection Surjection Bijection Some Interesting Stu ? C Counting The Pigeonhole Principle Composite Functions Inverses Graphs Relations Equivalence Relations Order Relations Polynomials Polynomial Arithmetic Polynomial Addition Polynomial Multipication Polynomial Division Prime Polynomials HCF and LCM Polynomial LCM The Remainder Theorem Roots of Polynomials Algebraic Numbers Counting Countable and uncountable sets Di ?erent In ?nities Foundations Basics Subgroups Sets and Venn Diagrams Mathematical Induction Remainder Theorem Pascal ? s Triangle Injections and Surjections Counting Samir Analysis True False Questions ??Give an Example ? Questions Toy Questions Conclusion C Introduction Pointless module Foundations is often considered by many ?rst year MORSE students as the most di ?cult module The aim of this guide is to try make it more fun and by fun I mean fun so that it becomes easier for you The notes will try create a link with modules materials that you will come across later in your ?rst year so that you don ? t feel that what you are doing is pointless Often how MORSE students feel about foundations determines how they feel about mathematics in general Many will be discouraged by this module while some will be inspired The best thing to do concerning Foundations is to keep in touch with the material by attempting and discussing the exercise sheets And yes unfortunately you ? ll have to study a bit over christmas for the January exams The pedantic mathematician You will ?nd during the course of your degree that mathematicians are the most pedantic academic creatures that you will ever come across The weirdest part of it all is how they set an injection between Mathematics and the set of arts Don ? t worry by the end of these notes you will get the joke During your ?rst year you will often ?nd yourselves proving obvious things like For any number prove that MA Any odd degree polynomial crosses the x-axis at least once MA You will also be proving more interesting