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Survey of Mathematical Ideas Math 100 Chapter 2 John Rosson Thursday January 25, 2007

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Basic Concepts of Set Theory 1.Symbols and Terminology 2.Venn Diagrams and Subsets 3.Set Operations and Cartesian Products 4.Cardinal Numbers and Surveys 5.Infinite Sets and Their Cardinalities

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Sets A set is a collection of objects. The objects in a set are called its elements or members. If A is a set and a is an element of A, we show this in symbols as follows: “a is an element of A” “a is not an element of A”

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Designating Sets Word description: Listing: Set builder: The empty set, designated Ø, is the set with no elements. The set of positive whole numbers which are less than 20 and evenly divisible by 7.

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Set Equality Set A is equal to set B if 1.every element of A is an element of B and 2.Every element of B is an element of A. This denoted, as one would expect “set A equals set B”“set A does not equal set B”

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Set Equality Examples:

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Sets of Numbers.

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Cardinality The cardinal number or cardinality of a set is the number of element in a set. In symbols, the cardinality of a set A is denoted Equal sets have equal cardinality, but sets with equal cardinality are not always equal. n(A)

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Cardinality Examples Intuitively, the sets in the above example are finite. On the other hand, the sets of numbers N, Z, Q and R are all examples of infinite sets. Later, we precise definitions of the words “finite” and “infinite” mean.

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Subsets Set A is a subset of set B if every element of A is also an element of B Denoted in symbols, “set A is a subset of set B”“set A is not a subset of set B” / If A and B are sets then A = B if A B and B A.

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Subsets Set A is a proper subset of set B if A B and A≠B. This denoted in symbols, “set A is a proper subset of set B”“set A is not a proper subset of set B”

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Subsets Examples. Let A be a set. /

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Sets can be elements. Any set can be an element of a set. If then

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Power Set The power set of set A, denoted is the set of all subsets of A. Thus

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Power Set Example In particular, the number of subsets of {1,2,3} is

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Power Set Theorem: The number of subsets of a finite set A is given by and the number of proper subsets is given by

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Power Set SetCardinality# Subsets# Proper Subsets Ø02 0 =11-1=0 {a}12 1 =22-1=1 {a,b}22 2 =44-1=3 {1,2,3}32 3 =87 {1,2,c,4,5}52 5 =3231 {1,2,3,…,100}1002 100 =12676 5060022822 9401496703 205376 12676506002 28229401496 703205375

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Assignments 2.3, 2.4, 2.5 Read Section 2.3 Due January 30 Exercises p. 73 1-6, 7-27, 47, 51, 52, 71, 75, 97, 115, 127, 129, 131, 133. Read Section 2.4 Due February 1 Exercises p. 79 1, 3, 5, 7, 9, 17, 19, 25, and 27. Read Section 2.5 Due February 6 Exercises p. 88 1-6, 7, 9, 11, 13, 14, 15, 24, 29, 32, 37, 38, 39, 40, 43.

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