Working WITH
Sets
Section 3-5
Goals
Goal
• To write sets and identify
subsets.
• To find the complement
of a set.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies the
goals to different and more
complex problems.
Vocabulary
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Set
Roster Form
Set-Builder Notation
Empty Set
Venn Diagram
Universal Set
Complement of a Set
Subset
Why Study Set Theory?
Understanding set theory helps people to …
• see things in terms of systems
• organize things into groups
• begin to understand logic
Set Concepts
Studying sets helps us categorize
information. It allows us to make sense of a
large amount of information by breaking it
down into smaller groups.
Sets:
•A set is a collection of objects.
–These objects can be anything: Letters, Shapes,
People, Numbers, Desks, cars, etc.
–Notation: Braces ‘{ }’, denote “The set of …”
•These objects are called elements or members
of the set.
•The symbol for element is  .
•For example, if you define the set as all the fruit
found in my refrigerator, then apple and orange
would be elements or members of that set.
Sets:
• Sets are inherently unordered:
– No matter what objects a, b, and c
denote,
{a, b, c} = {a, c, b} = {b, a, c} =
{b, c, a} = {c, a, b} = {c, b, a}.
• All elements are distinct (unequal);
multiple listings make no difference!
– {a, b, c} = {a, a, b, a, b, c, c, c, c}.
– This set contains at most 3 elements!
Sets:
• There are three methods used to
indicate a set:
1. Description
2. Roster form
3. Set-builder notation
• Venn Diagram - Used to display the
contents of a set and the
relationships between sets.
1. Description:
•Description means just that, words
describing what is included in a set.
•Example: “Set M is the set of months that
start with the letter J.”
2. Roster Form:
•Roster form lists all of the elements in the
set within braces {element 1, element 2, …}.
•Example: Set M = { January, June, July}
3. Set-Builder Notation:
•Set-builder notation is frequently used
in algebra.
•Example: M = { xx  is a month of the
year and x starts with the letter J}
•This is read, “Set M is the set of all the
elements x such that x is a month of the
year and x starts with the letter J”.
Set Summary:
•In summary the three methods used to
describe a set are:
1) Description: Set A is the integers 1, 2, 3, and 4.
2) Roster form: Set A = { 1, 2, 3, 4 }
3) Set-builder notation:
–A = { xx = 1, 2, 3, 4 }
Designating Sets
Sets are commonly given names (capital letters).
A = {1, 2, 3, 4}
The set containing no elements is called
the empty set (null set) and denoted
by { } or .
The Empty Set
• Any set that contains no elements
is called the empty set
• the empty set is a subset of every
set including itself
• notation: { } or 
Examples ~ both A and B are empty
A = {x | x is a Chevrolet Mustang}
B = {x | x is a positive number  0}
Set Notation Elements
• an element is a member of a set
• notation:
 means “is an element of”
 means “is not an element of”
• Examples:
– A = {1, 2, 3, 4}
1A 6A
2A zA
– B = {x | x is an even number  10}
2B 9B
4B zB
Example: Listing Elements of
Sets
Give a complete listing of all of the elements of the set
{x|x is a natural number between 3 and 8}
Solution
{4, 5, 6, 7}
When listing the elements of a set,
elements that occur more than once, are
not repeated when listing the elements in
set notation.
Set Theory Notation Summary
Symbol
Meaning
Upper case
Lower case
{ }
 or 
| or :
designates set name
designates set elements
enclose elements in set
is (or is not) an element of
such that (if a condition is true)
VENN DIAGRAMS
Venn diagrams are useful for presenting a
visual picture of set relationships.
Venn Diagrams
• Sets can be represented
graphically using Venn diagrams.
• In Venn diagrams:
–A rectangle represents the
universal set.
–Circles (and other geometric
figures) represents sets.
–Points (or words, nunbers)
represent elements.
Venn Diagrams
Subsets
•When working with a large group of
information, we often break it into smaller
sets called subsets.
Subsets of a Set
Set A is a subset of set B if every element
of A is also an element of B. In symbols
this is written A  B.
B
A
U
Subsets:
• Set A is a subset of set B, symbolized by A  B,
if and only if all the elements of set A are also
elements of set B. So to be a subset, all elements
of the set are also elements in another set (which
is either the same size or larger than the first set).
Subsets
• a subset part of or equal to another set
• notation:
 means “is a subset of”
 means “is not a subset of”
Subset Examples:
•
Given the sets A = { 1 , 2 }, B = { 1 , 2 ,
3 }, and D = { 1 , 2 , 3 }
1. A  B (said “A is a subset of B”) since
all A is in B. Note that this cannot be
written in reverse since B is not a
subset of A.
2. D  B since all D is in B.
Example: Subsets
Fill in the blank with  or 
 to make a true
statement.
a) {a, b, c} ___ { a, c, d}
b) {1, 2, 3, 4} ___ {1, 2, 3, 4}
Solution

a) {a, b, c} ___
 { a, c, d}
 {1, 2, 3, 4}
b) {1, 2, 3, 4} ___
Other Interesting Points About Subsets:
1. A  A (meaning - every set is a subset of
itself).
2. The empty set, , is a subset of every
set, including itself.
Number of Subsets
The number of subsets of a set with
n elements is 2n.
Example: Number of Subsets
Find the number of subsets of the
set {m, a, t, h, y}.
Solution
Since there are 5 elements, the number
of subsets is 25 = 32.
One Last Point:
•The number of distinct subsets of a
finite set A is 2n, where n is the
number of elements in set.
•Example: Given the set { S,L,E,D } .
The set has 4 elements, and 24 =
16. Thus, there are 16 distinct
subsets for that set (note that the
empty set is one of those 16 sets).
More Sets:
•Two more important sets to consider are
the empty set (also called null set) and the
universal set.
•The empty set is the set that contains no
elements. It is symbolized by { } or by .
•The universal set, symbolized by U, is
the set of all elements for any specific
discussion.
Universal Set
• The universal set is the set of all things
pertinent to a given discussion and is
designated by the symbol U
Example:
U = {all students at ATC}
Some Subsets:
A = {all HS students}
B = {freshmen students}
C = {sophomore students}
Universal Set Example
• the universal set is
a deck of ordinary
playing cards
• each card is an
element in the
universal set
• some subsets are:
–
–
–
–
face cards
numbered cards
suits
poker hands
Universal Set
In a venn diagram the rectangle
represents the universal set, U, and it
is required for all venn diagrams.
A
U
More on the Empty Set
• A set that has no elements is called the empty set
or null set.
• Yes, it is still considered a real set, even though it
has no elements.
• It is denoted by , or by { }.
• Since the empty set is a set, another set can
contain the empty set as one of its elements:
A ={, a}
This set has 2 elements
B = {}
This set has 1 element
C=
This set has 0 elements
Empty Set and Universal Set – Example:
•If we are given the universal set
•U = { Chris, Tom, Alex }, then only these
three names can be considered when
working with the problem.
•If A = { xx  U and x starts with the
letter J}, then our answer would be the
empty set (), since none of the names
in our universal set start with the letter J.
Complement of a Set:
•The complement of a set A, symbolized by A, is
all the elements in the universal set that are not in
A (everything outside of A).
One easy way to find this if the sets
are in roster form is to cross out each
element in U that is in set A. Then,
whatever is not crossed out in U, is
an element of A .
If U = { 1 , 2 , 3 , 4 , 5 , 6 }, and A = { 1 , 2 , 3 },
then A  = { 4
? ,}.5 , 6 }.
Complement of a Set:
• The complement of a set is the set of elements
which do not belong to the set being
complemented.
• Equivalent to the logic operation “not”
• Written as a prime, A’, or a superscripted ‘c’,
Ac.
• Example:
U = {a, b, c, d, e, u, v, w, x, y, z}
A = {a, b, c, x, y, z} and B = {a, b, c, d, e}
A’= {d, e, u, v, w} Bc = {u, v, w, x, y, z}
Complement of Sets: Venn
Diagrams
U = {a, b, c, d, e, u, v, w, x, y, z}
A = {a, b, c, x, y, z}
B = {d, e, y, z}
A’ = {d, e, u, v, w}
Joke Time
• What is the best state to buy a new soccer
uniform in?
• New Jersey
• Why is a football stadium always a cool
place to sit?
• It’s full of fans!
• What did the pony say when he had a cold?
• I’m just a little horse!
Assignment
3-5 Exercises Pg. 213 – 215: #10 – 56 even