Learning Objectives for Section 7.2 Sets
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Transcript of Learning Objectives for Section 7.2 Sets
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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}
In the Venn diagram on the left, the union of A and B is the entire region shaded.
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.
AA BB
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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.
AA BB 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:
{ is a rational number}
{ is a irrational number}
Q x x
I x x
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
}{' AxUxA
In the Venn diagram on the left, the rectangle represents the universal set.
A is the shaded area outside the set A.
U
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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?
A B
U
65 12 40
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2) 'n A
3) n A B
4) 'n A B
5) 'n A B
1) n U
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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
The number of elements in a set A is denoted by n(A), so in this case we are looking for
'N M
( ')n N M
The set N (news listeners) consists of
600 elements all together. The middle
part has _______, so the other part must
have _______ elements. Therefore,
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Solution(continued)
' ________ .n N M
Fill in the remaining blanks.
The study is based on 1000 commuters, so n(U)=___________.
______ listen to news but not music.
______ listen to music but not news
NM
U_____ people listen to neither news nor music
_______ listen to both music and news
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Examples From the Text
Page 364 # 2 – 42 even