L01 - Sets A B C Lecture1:Sep5 This Lecture

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Sets Lecture 1: Sep 5 A B C
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This Lecture We will first introduce some basic set theory before we do counting.  Basic Definitions  Operations on Sets  Set Identities  Russell’s Paradox
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Defining Sets We can define a set by directly listing all its elements. e.g. S = {2, 3, 5, 7, 11, 13, 17, 19},        S = {CSC1130, CSC2110, ERG2020, MAT2510} Definition:  A  set  is an unordered collection of objects. The objects in a set are called the  elements  or  members of the set S, and we say S  contains  its elements. After we define a set, the set is a single mathematical object, and it can be an element of another set. e.g. S = {{1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}
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Defining Sets by Properties ) { } ( | x A P x to define the set as the  set of elements , x,  in A  such that  x satisfies property P. It is inconvenient, and sometimes impossible,  to define a set by listing all its elements. Alternatively, we can define by a set by describing  the properties that its elements should satisfy. We use the notation e.g.
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Examples of Sets the set of all real numbers,          the set of all complex numbers,  the set of all integers,     the set of all positive integers              empty set ,              , the set with no elements. Well known sets: Other examples: The set of all polynomials with degree at most three: {1, x, x 2 , x 3 , 2x+3x 2 ,…}. The set of all n-bit strings: {000…0, 000…1, …, 111…1} The set of all triangles without an obtuse angle: {           ,           ,…   } The set of all graphs with four nodes: {             ,            ,           ,           ,…}
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Order number of occurence  are not important. e.g. {a,b,c} = {c,b,a} = {a,a,b,c,b} Membership x is an  element  of A x is  in  A 7     2/3    e.g. The most basic question in set theory is whether an element is in a set. Recall that Z is the set of all integers.  So             and                . Let P be the set of all prime numbers.  Then               and  Let Q be the set of all rational numbers.  Then               and  x is not an  element  of A x is not  in  A (will prove later)
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Size of a Set In this course we mostly focus on finite sets. e.g. if S = {2, 3, 5, 7, 11, 13, 17, 19}, then |S|=8.
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This note was uploaded on 11/11/2011 for the course MATH 112 taught by Professor Jarvis during the Winter '08 term at BYU.

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L01 - Sets A B C Lecture1:Sep5 This Lecture

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