Announcements
Finite Probability
Wednesday, August 31st
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MyMathLab 1 is due Wednesday Sept 7
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Problem Set 1 is due Friday Sept 9
Today: Sec. 5.1: Sets
Learn set theory notation
Understand set operations with Venn diagrams
Next Class: Sec. 5.2: Fundamental Principle of Counting
Cherveny
Aug 31
Math 1004: Probability
Sets
Definition
A set is a collection of objects, called the elements of the set.
We use brackets when giving the elements a set.
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List all the elements of the set
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Define the elements of a set by their properties
Cherveny
Aug 31
Math 1004: Probability
Describing a Set
Example
Let A be the set of all animals with four legs.
A = {cat, dog, elephant, . . . }
A = {x|x is an animal and x has four legs }
“A is the set of x such that x is an animal and x has four
legs”
Cherveny
Aug 31
Math 1004: Probability
Describing a Set
Example
Let B be the set of all even numbers from 4 to 16.
B = {4, 6, 8, 10, 12, 14, 16}
B = {x|x ∈ N and 4 ≤ x ≤ 16 and x is even}
Warning: Don’t repeat elements in a set. An object is either in the
set or it isn’t.
Cherveny
Aug 31
Math 1004: Probability
Intersection
Definition
The intersection of sets A and B, written A ∩ B, is the set of all
elements that A and B share in common.
Example
A = set of all months of the year that start with the letter J
B = set of all months it doesn’t snow in Boston
A ∩ B = {June, July}
Cherveny
Aug 31
Math 1004: Probability
Union
Definition
The union of sets A and B, written A ∪ B, is the set of all objects
that are in A or in B (or possibly both).
Example
C = set of all US states touching the Pacific Ocean
D = set of all US states that begin with the letter A
What is C ∪ D?
C ∪ D = {Hawaii, Alaska, California, Oregon, Washington,
Alabama, Arizona, Arkansas}
Note: Alaska only needs to be listed once.
Cherveny
Aug 31
Math 1004: Probability
Subsets
Definition
A set S is a subset of set A, written S ⊆ A, if every object of S is
also in A.
Example
Natural numbers:
N = {1, 2, 3, . . . }
Integers:
Z = {. . . , −2, −1, 0, 1, 2, . . . }
Every natural number is an integer, so
N⊆Z
“The naturals are a subset of the integers”
Cherveny
Aug 31
Math 1004: Probability
Empty Set
Definition
The empty set, written ∅, is the set with no objects:
∅ = {}
Example
If A is the set of all cartoon dogs and B is the set of planets in our
solar system, then..
A∩B =∅
Cherveny
Aug 31
Math 1004: Probability
Complement
Definition
The complement of a set, written A0 , is the set of all objects that
are not in A.
Note: Complement is relative to some “universe” set U of all
objects under consideration.
Example
Let A be the set of all movies starring Will Smith. What is A0 ?
Answer: Well.. depends. Is U the set of all movies? Is U the set
of everything Will Smith has released? Is it something else?
Tricky: What if U is the set of all movies starring Will Smith?
Answer: A0 = ∅.
Cherveny
Aug 31
Math 1004: Probability
Practice
Cherveny
Aug 31
Math 1004: Probability
Practice Answers
41. S 0
42. V ∩ C 0
43. (V ∪ C ) ∩ S 0
44. V 0 ∩ S 0 ∩ C 0 (another option is (V ∪ S ∪ C )0 )
45. C 0 ∩ V 0
Cherveny
Aug 31
Math 1004: Probability