Bridge_to_Abstract_Math_Lecture_Notes.pdf - BRIDGE TO...

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BRIDGE TO ABSTRACT MATHEMATICS LECTURE NOTES RAZVAN TEODORESCU Contents 1. Introduction 2 2. Set theory, functions, relations 3 2.1. Notation and conventions 3 2.2. Sets and elements 3 2.3. Operations with sets, comparing sets 3 2.4. Applications of set theory: Boolean algebras and mathematical logic 6 3. Algorithms: decidability, computability, optimality, induction 12 3.1. The principle of proof by mathematical induction 13 4. Proofs by contradiction 18 4.1. Topology 18 5. Algebraic structures: linear algebra 23 5.1. Models of proofs by reduction 26 5.2. Practice problems 31 6. Supplementary problems 32 Date : Fall 2016. 1
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2 BRIDGE TO ABSTRACT MATHEMATICS – LECTURE NOTES 1. Introduction This set of lecture notes consists of four main sections, to be covered sequentially. A brief overview of the focus for each section is given below. The first type of proof, by deduction 1 , uses direct reasoning starting from given facts, or premises . The accepted rules by which we can establish new facts starting from known facts are formalized under the name of mathematical logic . However, before defining mathematical logic, we need basic language to express mathematical statements, and this is the first reason to introduce elementary set theory . The second type of proof, known as mathematical induction , is an extension of the obvious idea of “proof by verification”, where there are only a finite number of concrete cases to consider. However, when the number of (verifiable) concrete instances that the statement can take is not finite, this algorithmic approach is not sufficient. Sometimes, the infinite number of instances can be counted , just like we count with natural numbers. Therefore, the problem of comparing sets (especially, infinite sets) arises, and through it, the notion of function (regarded as a way of comparing two sets). Another notion which arises at this point is that of equivalence relations , since two equinumerous sets are, in a certain sense, equivalent. Finally, when all the instances “to verify” form a countable set, and we have an algorithm that allows us to prove one instance using the previous ones, we arrive at the concept of mathematical induction. When the statement to prove does not reduce to a countable number of instances, it seems at first that we have no way to proceed. Luckily, principles of mathematical logic provide a direct way of proving some of these statements, through the notion of contradiction . No matter how complicated the statement might look like, we as- sume that it is true , and proceed to deductively derive other consequences it should have. If one of these consequences turns out to be false , however, then principles of logic dictate that the only possible interpretation of the apparent contradiction is that the original assumption was, in fact, unwarranted, and therefore its logical “opposite” (negation) must be true instead. Areas of mathematics where proofs by contradiction are encountered frequently are number theory and topology , which is the reason behind introducing some basic topological concepts in this section.
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