# Set Theory - homepages.ohiodominican.edu

```Thinking
Mathematically
Chapter 2 Set Theory
2.1 Basic Set Concepts
Basic Set Concepts
•A set is a collection of objects. Each object is
called an element of the set.
•A set must be well defined:
 Its contents can be clearly determined
 Its clear if an object is or is not a member of
the set.
Representing Sets
Word Description: Describe the set in
your own words, but be specific so the
elements are clearly defined.
Roster Method: List each element,
separated by commas, in braces.
Set-Builder Notation: {x | x is … word
description}.
The Set of Natural Numbers
N = {1,2,3,4,5,…}
 This is an example of a set
 We will be talking a lot more about sets of
numbers in Chapter 5
Examples: Representing Sets
Exercise Set 2.1 #3, 5, 13, 15, 25
• Well defined sets (T/F):
– The five worst U.S. presidents
– The natural numbers greater than one million
• Write a description for the set
{6, 7, 8, 9, …, 20}
• Express this set using the roster method:
The set of four seasons in a year.
{x | x  N and x > 5 }
The Empty Set
The empty set, also called the
null set, is the set that contains
no elements.
The empty set is represented by
{ } or Ø
Examples: Empty Sets
Exercise Set 2.1 #35, 37, 41, 45
Which sets are empty
• {x | x is a women who served as U.S. president
before 2000}
• {x | x is the number of women who served as
U.S. president before 2000}
• {x | x <2 and x > 5}
• {x | x is a number less that 2 or greater than 5}
The Notation  and 
The symbol  is used to indicate that
an object is an element of a set. The
symbol  is used to replace the
words “is an element of”
The symbol  is used to indicate that
an object is not an element of a set.
The symbol  is used to replace the
words “is not an element of”
Example: Set elements
Exercise Set 2.1 #51, 59, 63 (T/F)
• 5  { 2, 4, 6, …, 20}
• 13  {x | x  N and x < 13 }
• {3} {3, 4}
Definition of a Set’s Cardinal Number
The cardinal number of set A, represented
by n(A), is the number of distinct elements
in set A. The symbol n(A) is read “n of A”.
Repeated elements are not counted.
Exercise Set 2.1 #71
C = {x | x is a day of the week that begins with the letter A}
n( C) = ?
Definition of a Finite Set
Set A is a finite set if n(A) = 0 or n(A) is a
natural number. A set that is not finite is
called an infinite set.
Exercise Set 2.1 #91
{x | x  N and x >= 100}
Finite or infinite?
Definition of Equality of Sets
Set A is equal to set B means that set A and
set B contain exactly the same elements,
regardless of order or possible repetition of
elements. We symbolize the equality of sets
A and B using the statement A = B.
Definition of Equivalent Sets
Set A is equivalent to set B means that set
A and set B contain the same number of
elements. For equivalent sets, n(A) = n(B).
Exercise Set 2.1 #85
A = { 1, 1, 1, 2, 2, 3, 4}
B = {4, 3, 2, 1}
Are these sets equal?
Are these sets equivalent?
Thinking
Mathematically
Chapter 2 Set Theory
2.3 Venn Diagrams and Set Operations
[we’ll come back to 2.2]
Definition of a Universal Set
A universal set, symbolized by U, is a set
that contains all of the elements being
considered in a given discussion or problem.
Exercise Set 2.3 #3
A = {Pepsi, Sprite}
B = {Coca Cola, Seven-Up}
Describe a universal set that includes all elements in sets A and B
Venn Diagrams
“Disjoint” sets have no
elements in common.
U
A
U
A
All elements of B are
also elements of A.
The sets A and B have
some common elements.
B
B
U
A
B
Definition of the Complement of a Set
The complement of set A, symbolized by
A´, is the set of all elements in the
universal set that are not in A.
This idea can be expressed in set-builder
notation as follows:
A´ = {x | x  U and x  A }.
Complement of a Set
U
A
A’
Example: Set Complement
Exercise Set 2.3 #11
U = {1, 2, 3,…, 20}
A = {1, 2, 3, 4, 5}
B = {6, 7, 8, 9}
C = {1, 3, 5, …, 19}
D = {2, 4, 6, …, 20}
´=?
C
Definition of Intersection of Sets
The intersection of sets A and B, written
AB, is the set of elements common to
both set A and set B. This definition can be
expressed in set builder notation as follows:
A  B = { x | x  A AND x  B}
U
A
B
Definition of the Union of Sets
The union of sets A and B, written A  B,
is the set of elements that are members of
set A or of set B or of both sets. This
definition can be expressed in set-builder
notation as follows:
A  B = {x | x  A OR x  B}
U
A
B
The Empty Set in Intersection and
Union
For any set A:
1. A ∩  = 
2. A   = A
Examples: Union / Intersection
Exercise Set 2.3 #17, 19, 33, 35
U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}
B = {1, 2, 3}
C = {2, 3, 4, 5, 6}
• AB=?
• AB=?
• A=?
• A∩ =?
Cardinal Number of the Union of
Two Sets
n(A U B) = n(A) + n(B) – n(A ∩B)
Exercise Set 2.3 #93
–
–
–
–
Set A 17 elements
Set B 20 elements
There are 6 elements common to the two sets
How many elements in the union?
Thinking
Mathematically
Chapter 2 Set Theory
2.2 Subsets
Definition of a Subset of a Set
Set B is a subset of set A, expressed as
BA
if every element in set B is also an element in set A.
U
A
B
Every set is a subset of itself: A  A
Definition of a Proper Subset of a Set
Set B is a proper subset of set A, expressed
as B  A, if set B is a subset of set A and
sets A and B are not equal ( A  B ).
What is an improper subset?
The Empty Set as a Subset
1. For any set B,   B.
2. For any set B other than the empty set,
  B.
Example: Subsets
• Exercise Set 2.2 #3, 45, 43, 47
• {-3, 0, 3} ____ {-3, -1, 1, 3}
• (, , both, neither)
• {Ralph}  {Ralph, Alice, Trixie, Norton}
(T/F)
• Ralph  {Ralph, Alice, Trixie, Norton} (T/F)
•   {Archie, Edith, Mike, Gloria} (T/F)
Thinking
Mathematically
Chapter 2 Set Theory
2.4 Set Operations and Venn Diagrams
With Three Sets
Example: Operations with three sets
Exercise Set 2.4 #3, 15
• U = {1, 2, 3, 4, 5, 6, 7} • U = {a, b, c, d, e, f, g, h}
A = {1, 3, 5, 7}
B = {1, 2, 3}
C = {2, 3, 4, 5, 6}
(A  B) ∩ (A  C)
A = {a, g, h}
B = {b, h, h}
C = {b, c, d, e, f}
(A  B) ∩ (A  C)
Example – Venn Diagrams
Exercise Set 2.4 #35, 37
U
A
B
4,5
1, 2, 3
7, 8
10, 11
6
9
12
13
AB=?
C
(A  B)’ = ?
Example – Venn Diagrams
Exercise Set 2.4 #27, 29
U
A
I
II
B
III
IV V VI
VII
C
AC=?
A∩B=?
De Morgan’s Laws
(using Venn Diagrams as a proof)
• (A U B)' = A' ∩ B': The complement of the
union of two sets is the intersection of the
complement of those sets.
U
B
A
U
A
B
U
A
B
De Morgan’s Laws
• (A ∩ B)' = A' U B': The complement of the
intersection of two sets is the union of the
complement of those sets.
U
U
A
B
B
A
U
A
B
Examples: DeMorgan’s Laws
U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}
B = {1, 2, 3}
• (A ∩ B) ' = ?
• A'UB'=?
Thinking
Mathematically
Chapter 2 Set Theory
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