Transcript Set Notation

Aim: How do we deal with chaos?
Do Now:
How many of you drive yourself to school?
How many of you are driven to school?
How many of you take the bus to school?
How many of you walk to school?
How many of you get to school by other
means?
Aim: Set Notation
Course: Math Literacy
Terminology of sets
Sets deal with sorting of objects into similar
groupings, allowing us to order and structure the
world. Our minds cannot find order and meaning
without creating collections.
•A set is a collection of objects whose contents
can be clearly determined.
•Objects in sets are called members or elements.
ex. set of days of week whose elements include
Monday, Tuesday, Wednesday, Thursday, Friday,
Saturday and Sunday.
Aim: Set Notation
Course: Math Literacy
Terminology of sets
Three methods commonly used to designate a set.
word description
The set of the days of the week are the elements
Monday, Tuesday, Wednesday, Thursday, Friday,
Saturday and Sunday.
roster method
W = {Monday, Tuesday, Wednesday, Thursday,
Friday, Saturday, Sunday} order not important
only brackets and commas
Aim: Set Notation
Course: Math Literacy
Terminology of sets
Three methods commonly used to designate a set.
set builder notation
W = {x | x is a day of the week}
Set of
W
is
the
set of
all
elements
x
such
that
“Set W is the set of all elements x such
that x is a day of the week.”
Aim: Set Notation
Course: Math Literacy
Model Problem
Express set A = {x | x is a month that begins with
the letter M} using the roster method.
A = {March, May}
Express set A = {x | x is a month that begins with
the letter R} using the roster method.
A = { } or 
Empty Set – also called the null set, is the set
that contains no elements.
Aim: Set Notation
Note: {} is a set
Course: Math Literacy
containing the
element 
Terminology of sets
well defined – a set must be well defined,
meaning the contents are clearly determined
well defined
ex. the collection of actors who have won
Academy Awards is a set.
not well defined
ex. the collection of actors great actors.
subject to interpretation
Aim: Set Notation
Course: Math Literacy
Terminology of sets
The symbol  is used to indicate that an object
is an element of a set. It is used to replace the
words, “is an element of.”
r  {a, b, c, . . . , z}
ellipsis
elements of set continue
in same manner up to
and including the z.
The symbol  is used to indicate that an object
is not an element of a set. It is used to replace
the words, “is not an element of.”
7  {1, 2, 3, 4,
5}
Aim: Set Notation
Course: Math Literacy
Model Problem
True or False?
8  {1, 2, 3, . . . ,
10}
r  {a, b, c, z}
T
T
{Monday}  {x | x is a day of the week} T
Aim: Set Notation
Course: Math Literacy
Terminology of sets
N = {1, 2, 3, 4, 5, . . .} is the set of Natural Numbers.
Express the following sets using the roster method.
Set A is the set of natural numbers less than 5.
A = {1, 2, 3, 4}
B is the set of natural numbers greater than or
equal to 25.
A = {25, 26, 27, 28, . . .}
E = {x|x  N and x is even}
E = {2, 4, 6, 8, . . .}
F = {x|x  N and x is between 6 and 10}
Course: Math Literacy
Aim: Set Notation
F = {7, 8, 9}
Terminology of sets
Cardinal Number – the number of elements in a
set. Also called the cardinality of the set.
The cardinal number of set, represented by n(A), is
the number of distinct elements is set A. The
symbol n(A) is read “n of A.”
ex. Z = {a, e, i, o, u}
set Z has a cardinality of 5 or n(Z) = 5
Find the cardinal number of the following sets.
A = {7, 9, 11, 13} n(A) = 4
B = {0} n(B) = 1
C = {13, 14, 15, . . . , 22, 23}
Aim: Set Notation
n(C) = 11
 n() = 0
Course: Math Literacy
Terminology of sets
Finite Set – if n(A) = 0 or n(A) is a natural number.
A set that is not finite is called an infinite set.
{3, 6, 9, 12, . . . } is an infinite set
and has no cardinality
Equal Sets – contain exactly the same elements,
regardless of order or possible repetition of
elements. Equality of sets is symbolized as A = B
A = {w, x, y, z} and B = {z, y, x, w}
A=B
n(A) = n(B)
Aim: Set Notation
Course: Math Literacy
Terminology of sets
Equivalent Sets – contain the same number of
elements. Cardinalities are equal: n(A) = n(B)
Equal, equivalent, or both?
A = {The Graduate, The Godfather, Titanic} B
= {Hoffman, Brando, DiCaprio}
Not equal – do not contain exactly the same elements
Equivalent – each contains three elements:
n(A) = n(B)
note: repeating elements in a set do not add new
elements to the set.
Math Literacy
A = {1, 1, 2,Aim:2,Set3}Notation
and B = {1, 2, 3}; A =Course:
B; n(A)
= n(B)
Terminology of sets
Subsets – A is a subset of set B, expressed as A  B
if every element of set A is also an element of set B.
 means ‘is not a subset’
Let A = {x|x is a planet in the Earth’s solar system.
B = {Mercury, Venus, Earth, Mars, Jupiter}
C = {1 Ceres}
BA
CA
1 Ceres is the solar system’s largest asteroid. It is
974 km in Aim:
diameter
25%
of the
Course:
Math Literacy
Set Notation and contains about
mass of all the asteroids combined.
Relationships between 2 Sets
Proper Subset – Set A is a proper subset of set B,
expressed as A  B, if set A is a subset of set B and
sets A and B are not equal (A  B)
A B
Study Tip
A is a subset of B
a<b
a is less than or
equal to b
A B
a<b
A is a proper
subset of B
a is less than b but
not equal to b
Aim: Set Notation
Course: Math Literacy
Model Problems
Because every twentieth century US president was
male, this means that the set of twentieth century
US presidents is a subset of males.
{x | x is a 20th century US President}  {x | x is a male
Write  or  in each blank
A = {1, 3, 5, 7}
 B
B = {1, 3, 5, 7, 9, 11} A ____
A = {set of music written in 20th century}
 B
B = {set of jazz music} A ____
Aim: Set Notation
Course: Math Literacy
Number of Subsets
1. For any set B,   B.
2. For any set B other than the empty set,   B.
Number of
Number of
Set
List of All Subsets
Elements
Subsets
{ }
0
{ }
1
{a}
1
{ }{a}
2
{ a, b }
2
{ } { a } { b } { a, b }
4
{ }{a}{b}{c}
{ a, b, c }
3
{ a, b } { b, c } { a, c }
8
{ a, b, c }
n.
The number
of
subsets
of
a
set
with
n
elements
is
2
Course: Math Literacy
Aim: Set Notation
The number of proper subsets of a set is 2n – 1.
Model Problem
You recently purchase three books:
{The Color Purple, Hannibal, The Royals}
You are deciding which books, if any to take on
vacation. You have enough room to pack up to
three books,
but may
fewern or
not at all.
The number
of subsets
of atake
set with
elements
is 2n.
a. Find the number of subsets of the given set.
n = 3; 23 = 8
b. List all the subsets.
c. How many of the subsets are proper subsets?
The number of proper subsets of a set is 2n – 1.
Aim: Set Notation
3–1=7
2
Course: Math Literacy
Aim: Set Notation
Course: Math Literacy