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

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

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”

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.

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”

Set Equality Examples:

Sets of Numbers.

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)

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.

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.

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”

Subsets Examples. Let A be a set. /

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

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 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} =

Assignments 2.3, 2.4, 2.5 Read Section 2.3 Due January 30 Exercises p , 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 , 7, 9, 11, 13, 14, 15, 24, 29, 32, 37, 38, 39, 40, 43.