Download Chapter 1 Sets and Counting

Chapter 1 Sets and Counting – Lecture
Homework: Exercises and Review at end of chapter
 A set is a collection of objects.
o No repeated objects.
o Set is well-defined (It is easy to determine if an object
belongs to the set)
 {x | x is a real number} is well defined.
 {x | x plays basketball well} is not well defined.
o {x | x  0} is read the set of all x such that x is less than or
equal to 0.
 A member of the set is called an element
o 5  {Integers} read 5 is an element of the set of integers.
o 6  {1,2,3} read 6 is not an element of the set 1, 2, 3.
 Worksheet: Part 1
 Describing Sets
o Roster – list all the elements {ford, chevy, Chrysler}
o Set builder notation – {x | x is a state in the United States}
 Worksheet: Part 2
 Two sets are equal if they have the same elements.
 Two sets are equivalent if they have the same number of
elements.
o {1, 3, 2} and {1, 2, 3} are equal.
o {1, 2, 3} and {a, b, c} are not equal but equivalent.
o Notice: The {x | x is an integer} and {x | x is an even
integer} are not equal, but they are equivalent. (why?)
 Worksheet: part 3
 The following are special sets:
o Ø, or {} – the empty set that contains no elements
o N = {1, 2, 3, 4, …} the natural numbers
o Z = {…, -3, -2, -1, 0, 1, 2, …} the integers
o
o R = {all real numbers}
 A set can contain a set as an element.
o {1, 2, 3, Ø, {3}} is a set containing 5 elements. Note that
3, and {3} are not the same.
 Cardinality – The number of elements in a set.
o The cardinality of {a, b, c} is 3.
o The cardinality of Ø is 0.
o 1-1 Correspondence – Two sets A and B have a 1-1
correspondence if every element in A can be paired with
one element in B, and every element in B can be paired
with one element in A.
The positive integers and the even positive integers have a 1-1
correspondence.
 Worksheet: part 4
 Subsets; Counting
o B is a subset of A if every element in B is in A.
o Counting the number of subsets.
 example: {1} – two subsets (don’t select any
elements, select one element)
 example: {1, 2} – four
 example: {1, 2, 3} – 8
 Worksheet: part 5:
 Theorem: A set with n elements has 2n subsets.
o Note: for Ø, 20 = 1
 Proper and improper subsets
o Every set is a subset of itself – improper.
o A proper subset is a subset that is not equal to the set:
o Note: {2, 3, 5} has 7 proper subsets 23 – 1
o Definition of an infinite set: A set is infinite if it can be
put into a 1-1 correspondence with a proper subset of
itself.
 Worksheet: part 6
 Fundamental counting principle
o Suppose Jane has two jackets and three hats, and she is
trying to decide what jacket and hat to wear. How many
choices does she have altogether?
 The fundamental counting principle says:
o If there are m ways to do task A
o If there are n ways to do task B
o Then there are mn ways to do both tasks.
o This idea extends to any number of tasks.
 Count the number of ways to do each task.
Multiply the counts to obtain the total number
of ways to perform the tasks.
 Worksheet: part 7