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**Survey of Mathematical Ideas Math 100 Chapter 2**

John Rosson Tuesday January 30, 2007

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**Basic Concepts of Set Theory**

Symbols and Terminology Venn Diagrams and Subsets Set Operations and Cartesian Products Cardinal Numbers and Surveys Infinite Sets and Their Cardinalities

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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 Ø 20=1 1-1=0 {a} 1 21=2 2-1=1 {a,b} 2 22=4 4-1=3 {1,2,3} 3**

Cardinality # Subsets # Proper Subsets Ø 20=1 1-1=0 {a} 1 21=2 2-1=1 {a,b} 2 22=4 4-1=3 {1,2,3} 3 23=8 7 {1,2,c,4,5} 5 25=32 31 {1,2,3,…,100} 100 2100=

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Complement The collection of all possible element of sets, either stated or implied, is called the universal set, often denoted U. For any subset A of the universal set U, the complement of A, denoted is the set elements of U not in A. Thus

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Complement Examples. Let U={a,b,c,d,e,f,g,h,i,j,k,l}, A={a,b,c,d}. Let C D.

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Venn Diagrams A Venn diagram is a pictorial representation of sets and their various relations and operation. The first picture below represents the universal set U, a set A, and the complement of A. The second represents the relation

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Numbers as Sets All mathematical objects can be defined in terms of sets. The example below indicates how one might define the first five whole numbers as sets.

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**Intersection The intersection of sets A and B, denoted**

is the set of elements common to both A and B.

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Intersection Let A and B be sets with A B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set Sets with empty intersection are called disjoint. Thus, every set is disjoint from its complement.

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**Union The union of sets A and B, denoted**

is the set of elements belonging to either of the sets.

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Union Let A and B be sets with A B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set

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**Difference The difference of sets A and B, denoted**

is the set of elements belonging to set A but not to set B.

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Difference Let A and B be sets with A B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set

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Ordered Pairs The ordered pair of with first component a and second component b, denoted is defined to be the set Thus, Note that for ordered pairs, order is important. So In particular,

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**Cartesian Product The Cartesian product of sets A and B, denoted**

is the set

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**Cartesian Product Let C = {1,2,3} and D = {3,a,b}.**

In general, for sets A and B: So in the example

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Operations on Sets Operations on sets can be combined. Let A={a,b}, B={b,c}, C={c,d}, D={b,d} and E={a,c}. Calculate in list form. Working from the inside out

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Venn Diagrams Here is the previous set calculation as a Venn diagram. The is no adequate Venn diagram for the Cartesian product. a b c A B b c d C c b d D

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De Morgan’s Laws For and sets A and B, the complement of their intersection is the union of their complements, and the complement of their unions is the intersection of their complements.

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De Morgan’s Laws The set will be all the blue not in A and not in B.

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**Assignments 2.4, 2.5, 3.1 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. Read Section 3.1 Due February 8 Exercises p. 99 1-9, 39-47, 49-53, 57-74

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