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Unformatted text preview: Discrete Mathematics and Proofs: an introduction Periklis A. Papakonstantinou ? University of Toronto In this handout we (i) briefly mention the role of Formal Logic in this course, (ii) we prove that every subset of an upper bounded set is also bounded and (iii) we prove that among every 6 people either there are three mutual acquaintances or there are three mutual strangers (or both). Mathematics, Formal Logic, Discrete Mathematics and Proofs During your undergraduate engineering career you will come across with complicated for- mulas drawn from several mathematical disciplines. A mere, blind application of such a formula is not what we will call as mathematics. Every field of mathematics is indispens- able from the notion of proof. For example, Calculus is mathematics. The developments behind the notion of continuity, integration etc. involve reasoning and arguments and thus Calculus is certainly an important area of mathematics. On the other hand a direct, me- chanical application of some ready-to-use tools from Calculus, e.g. in the direct computation of a definite integral, does not qualify as mathematics. Maintaining the analogies, would you ever call | 1- 2 | = 1 as mathematics? Learn how to do and read proofs for statements related to Discrete Mathematics is the main subject of this course. Most of the basic de- velopments in Discrete Mathematics are simple, they can be quickly understood, they are very intuitive, elegant and fun. As a bonus you will see that Discrete Mathematics have a handful of applications in many areas of Electrical and Computer Engineering. Every discipline has its own language. Logic is the language of mathematics. In a first course in discrete math students begin by being introduced to elements of propositional logic and first-order logic. Although Logic has an autonomous existence as a field we will not get into this in ECE190. Here, as far as it concerns Formal Logic, students are simply asked to perform mechanical tasks, playing around with truth assignments, constructing truth tables of propositional expressions like ( P Q ) etc. As it turns out this mechanical understanding of Logic is very helpful, albeit the following important warning. When people start delving into the mechanics of basic formal logic they tend to loose the big picture. For the purpose of this course all these things from Formal Logic enhance our ability to write and read proofs written in English. My advice is whatever you do mechanically with Formal Logic try to translate it to (associate it with) reasoning and arguments inside your English-written proofs. ? This document possibly contains typos. Please submit any typos, remarks, suggestions to [email protected] . 2 Periklis Papakonstantinou (ECE 190, Fall 2006) An introductory example A definition is something that historically, in most cases, came to life out of examples. We do one example and we observe something interesting. Then, we do another and we alsodo one example and we observe something interesting....
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- Fall '06
- Logic, Model theory, independent set