138-16-FALL-WEEK2-Sept.19,23-H1.pdf - UNIVERSITY OF TORONTO...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 WEEK #2 (SEPTEMBER 19 AND 23) Part 1. The language of “Sets”. - Sets and Elements: Frequently, we may want to consider a number of distinct objects as a single entity. In such cases, we will refer to the entity as a “Set” and we will refer to the distinct objects as the “Elements” of the set. Most of the sets that we will consider in our course will be sets whose elements are numbers or sets whose elements are other math-related objects, like functions, geometric figures, etc. - Describing sets: As an example, suppose that we want to use 𝑆𝑆 to denote the set consisting of the numbers 6 , 8 , 10 and 12 . Below are the two symbolic forms that we will most commonly use to describe sets like 𝑆𝑆 . a) By “listing its elements”: 𝑆𝑆 = {6,8,10,12}. b) By “stating a property that specifies its elements”: 𝑆𝑆 = { 𝑥𝑥 | 𝑥𝑥 is an even integer between 5 and 13} . Note: The symbol “ | ” above is a common abbreviation for the words “such that”. - Some useful extensions to the notion of set: a) The set 𝑆𝑆 = { } (the set containing no elements). { } is usually called “the empty set” and is denoted 𝜙𝜙 . b) Sets like 𝐴𝐴 = {7} , 𝐵𝐵 = �− 5 9 and 𝐶𝐶 = {0} (sets containing exactly one element). Note that now the expressions 7 and {7} represent to two totally different mathematical entities. c) Sets like 𝐷𝐷 = { 𝑥𝑥 | 𝑥𝑥 is a rational number less than 8 3 } (sets containing infinitely many elements). d) Sets like 𝐸𝐸 = 2,3,7, {3,5}, {1,3,6,7,9} such that some (or all) of their elements are themselves sets. - Some useful notations and terminology: a) If 𝑥𝑥 is an element of a set 𝑆𝑆 , then we write 𝑥𝑥 𝑆𝑆 and we usually say that “ 𝑥𝑥 belongs to 𝑆𝑆 ”. Similarly, 𝑥𝑥 𝑆𝑆 means that 𝑥𝑥 is not an element of 𝑆𝑆 and we say that “ 𝑥𝑥 does not belong to 𝑆𝑆 ”. b) Two sets 𝐴𝐴 and 𝐵𝐵 are said to be “equal”, written 𝐴𝐴 = 𝐵𝐵 , when they have precisely the same elements.
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