Survey of Mathematical Ideas Math 100 Chapter 2 John Rosson Tuesday January 30, 2007

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

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

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

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

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=1267650600228229401496703205376 1267650600228229401496703205375

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

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

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

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.

Intersection The intersection of sets A and B, denoted is the set of elements common to both A and B.

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.

Union The union of sets A and B, denoted is the set of elements belonging to either of the sets.

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

Difference The difference of sets A and B, denoted is the set of elements belonging to set A but not to set B.

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

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,

Cartesian Product The Cartesian product of sets A and B, denoted is the set

Cartesian Product Let C = {1,2,3} and D = {3,a,b}. In general, for sets A and B: So in the example

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

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

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.

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

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