Transcript 7.2

Learning Objectives for Section 7.2
Sets
After today’s lesson, you should be able to
 Identify and use set properties and set notation.
 Perform set operations.
 Solve applications involving sets.
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Set Properties and Set Notation
 Definition: A set is any collection of objects
 Notation:
eA
means “e is an element of A”,
or “e belongs to set A”.
eA
means “e is not an element of A”.
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Set Notation (continued)
AB
means “A is a subset of B”
A=B
means “A and B have exactly the same elements”
AB
means “A is not a subset of B”
AB
means “A and B do not have exactly the same elements”
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Set Properties and Set Notation
(continued)
Example of a set: Let A be the set of all the letters in the alphabet.
We write that as
A = { a, b, c, d, e, …, z}.
This is called the listing method of specifying a set.
 We use capital letters to represent sets.
 We list the elements of the set within braces, separated by
commas.
 The three dots (…) indicate that the pattern continues.
Question: Is 3 a member of the set A?
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Set-Builder Notation
Sometimes it is convenient to represent sets using set-builder
notation.
Example: Using set-builder notation, write the letters of the
alphabet.
A = {x | x is a letter of the English alphabet}
This is read, “the set of all x such that x is a letter of the
English alphabet.”
It is equivalent to A = {a , b, c, d, e, …, z}
Note: {x | x2 = 9} = {3, -3}
This is read as “the set of all x such that the square of x
equals 9.”
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Null Set
Example: What are the real number solutions of the equation
x2 + 1 = 0?
Answer: ________________________________________
________________________________________________
We represent the solution as the __________, written ____ or ___.
It is also called the _______________ set.
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Subsets
A is a subset of B if every element of A is also contained in B.
This is written
A  B.
For example, the set of integers
{ …-3, -2, -1, 0, 1, 2, 3, …}
is a subset of the set of real numbers.
Formal Definition:
A  B means “if x  A, then x  B.”
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Subsets
(continued)
Note:
Every set is a subset of itself.
Ø (the null set) is a subset of every set.
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Number of Subsets
Example: List all the subsets of set A = {bird, cat, dog}
For convenience, we will use the notation A = {b, c, d} to
represent set A.
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Union of Sets
(OR)
A  B = { x | x  A or x  B}
The union of two sets A and B is the set of all elements formed
by combining all the elements of set A and all the elements of set
B into one set. It is written A  B.
A
B
In the Venn diagram on the left,
the union of A and B is the entire
region shaded.
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Intersection of Sets
(AND)
A  B = { x | x  A and x  B}
The intersection of two sets A and B is the set of all elements
that are common to both A and B. It is written A  B.
A
B
In the Venn diagram on the left,
the intersection of A and B is the
shaded region.
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Example
Example: Given A = {3, 6, 9, 12, 15} and B = {1, 4, 9, 16}
find:
a) A  B .
b) A  B.
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Disjoint Sets
If two sets have no elements in common, they are said to be
disjoint. Two sets A and B are disjoint if
A  B = .
Example: The rational and irrational numbers are disjoint.
In symbols: Q  { x x is a rational num ber}
I  {x
x is a irrational num ber}
QI 
QI 
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The Universal Set
The set of all elements under consideration is called the
universal set U.
U
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The Complement of a Set
(NOT)
The complement of a set A is defined as the set of elements that
are contained in U, the universal set, but not contained in set
A. The symbolism and notation for the complement of set A are
A '  {x  U
x  A}
U
In the Venn diagram on the left, the
rectangle represents the universal set.
A is the shaded area outside the set A.
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Venn Diagram
Refer to the Venn diagram below. The indicated values represent
the number of elements in each region. How many elements
are in each of the indicated sets?
1) n  U 
U
A
B
65
12
40
25
2)
n  A '
3)
nA  B
4)
n  A ' B 
5)
n   A  B  '
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Application*
A marketing survey of 1,000 car commuters found that 600
listen to the news, 500 listen to music, and 300 listen to
both.
Let N = set of commuters in the sample who listen to news
Let M = set of commuters in the sample who listen to music
Find the number of commuters in the set N  M '
The number of elements in a set A is denoted by n(A), so in
this case we are looking for n ( N  M ')
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Solution
(continued)
The study is based on 1000 commuters, so n(U)=___________.
U
_____ people listen to neither
news nor music
M
N
______
listen to
news but not
music.
______
listen to
music
but not
news
The set N (news listeners) consists of
600 elements all together. The middle
part has _______, so the other part must
have _______ elements. Therefore,
n  N  M '   ________ .
Fill in the remaining blanks.
_______ listen to both
music and news
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Examples From the Text
 Page 364 # 2 – 42 even
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