university-logo Set Properties and Set Notation Set Operations Math 1300 Finite Mathematics Section 7-2: Sets Department of Mathematics University of Missouri October 16, 2009 Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Math 1300 Finite MathematicsSection 7-2: Sets
Department of MathematicsUniversity of Missouri
October 16, 2009
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
A set is any collection of objects specified in such a waythat we can determine whether a given object is or is not inthe collection.
Capital letters, such as A, B, and C are often used todesignate particular sets.Each object in a set is called a member or element of theset.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
A set is any collection of objects specified in such a waythat we can determine whether a given object is or is not inthe collection.Capital letters, such as A, B, and C are often used todesignate particular sets.
Each object in a set is called a member or element of theset.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
A set is any collection of objects specified in such a waythat we can determine whether a given object is or is not inthe collection.Capital letters, such as A, B, and C are often used todesignate particular sets.Each object in a set is called a member or element of theset.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Symbolically,
a ∈ A means "a is an element of set A"a 6∈ A means "a is not an element of set A"
A set without any elements is called the empty, or null,set.Symbollically
∅ denotes the empty set
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Symbolically,
a ∈ A means "a is an element of set A"a 6∈ A means "a is not an element of set A"
A set without any elements is called the empty, or null,set.
Symbollically
∅ denotes the empty set
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Symbolically,
a ∈ A means "a is an element of set A"a 6∈ A means "a is not an element of set A"
A set without any elements is called the empty, or null,set.Symbollically
∅ denotes the empty set
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
A set is usually described either by listing all its elementsbetween braces { } or by enclosing a rule within braces thatdetermines the elements of the set.
Thus, if P(x) is a statement about x , then
S = {x |P(x)} means “S is the set of all x such that P(x) is true”
Example:Rule Listing
{x |x is a weekend day} = {Saturday, Sunday}{x |x2 = 4} = {−2, 2}
{x |x is a positive odd number} = {1, 3, 5, . . . }
The first two sets in this example are finite sets; the last set isan infinite set.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
A set is usually described either by listing all its elementsbetween braces { } or by enclosing a rule within braces thatdetermines the elements of the set.
Thus, if P(x) is a statement about x , then
S = {x |P(x)} means “S is the set of all x such that P(x) is true”
Example:Rule Listing
{x |x is a weekend day} = {Saturday, Sunday}{x |x2 = 4} = {−2, 2}
{x |x is a positive odd number} = {1, 3, 5, . . . }
The first two sets in this example are finite sets; the last set isan infinite set.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
A set is usually described either by listing all its elementsbetween braces { } or by enclosing a rule within braces thatdetermines the elements of the set.
Thus, if P(x) is a statement about x , then
S = {x |P(x)} means “S is the set of all x such that P(x) is true”
Example:Rule Listing
{x |x is a weekend day} = {Saturday, Sunday}{x |x2 = 4} = {−2, 2}
{x |x is a positive odd number} = {1, 3, 5, . . . }
The first two sets in this example are finite sets; the last set isan infinite set.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
A set is usually described either by listing all its elementsbetween braces { } or by enclosing a rule within braces thatdetermines the elements of the set.
Thus, if P(x) is a statement about x , then
S = {x |P(x)} means “S is the set of all x such that P(x) is true”
Example:Rule Listing
{x |x is a weekend day} = {Saturday, Sunday}{x |x2 = 4} = {−2, 2}
{x |x is a positive odd number} = {1, 3, 5, . . . }
The first two sets in this example are finite sets; the last set isan infinite set.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.
A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”∅ is a subset of every set.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.
If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”∅ is a subset of every set.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.
Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”∅ is a subset of every set.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”
∅ is a subset of every set.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
When listing the elements of a set, we do not list anelement more than once, and the order in which theelements are listed does not matter.A is a subset of B if every element of A is also an elementof B.If A and B have exactly the same elments, then the twosets are said to be equal.Symbollically,A ⊂ B means “A is a subset of B”A = B means “A equals B”A 6⊂ B means “A is not a subset of B”A 6= B means “A and B do not have the same elements”∅ is a subset of every set.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: List all subsets of the set A = {b, c, d}.
First, ∅ ⊂ A.
Also, every set is a subset of itself, so A ⊂ A.
What one-element subsets? {b}, {c} and {d}.
What two-element subsets are there? {b, c}, {b, d}, {c, d}
There are a total of 8 subsets of set A.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: List all subsets of the set A = {b, c, d}.
First, ∅ ⊂ A.
Also, every set is a subset of itself, so A ⊂ A.
What one-element subsets? {b}, {c} and {d}.
What two-element subsets are there? {b, c}, {b, d}, {c, d}
There are a total of 8 subsets of set A.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
Example: List all subsets of the set A = {b, c, d}.
First, ∅ ⊂ A.
Also, every set is a subset of itself, so A ⊂ A.
What one-element subsets? {b}, {c} and {d}.
What two-element subsets are there? {b, c}, {b, d}, {c, d}
There are a total of 8 subsets of set A.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
Example: List all subsets of the set A = {b, c, d}.
First, ∅ ⊂ A.
Also, every set is a subset of itself, so A ⊂ A.
What one-element subsets?
{b}, {c} and {d}.
What two-element subsets are there? {b, c}, {b, d}, {c, d}
There are a total of 8 subsets of set A.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
Example: List all subsets of the set A = {b, c, d}.
First, ∅ ⊂ A.
Also, every set is a subset of itself, so A ⊂ A.
What one-element subsets? {b}, {c} and {d}.
What two-element subsets are there? {b, c}, {b, d}, {c, d}
There are a total of 8 subsets of set A.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
Example: List all subsets of the set A = {b, c, d}.
First, ∅ ⊂ A.
Also, every set is a subset of itself, so A ⊂ A.
What one-element subsets? {b}, {c} and {d}.
What two-element subsets are there?
{b, c}, {b, d}, {c, d}
There are a total of 8 subsets of set A.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
Example: List all subsets of the set A = {b, c, d}.
First, ∅ ⊂ A.
Also, every set is a subset of itself, so A ⊂ A.
What one-element subsets? {b}, {c} and {d}.
What two-element subsets are there? {b, c}, {b, d}, {c, d}
There are a total of 8 subsets of set A.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
Example: List all subsets of the set A = {b, c, d}.
First, ∅ ⊂ A.
Also, every set is a subset of itself, so A ⊂ A.
What one-element subsets? {b}, {c} and {d}.
What two-element subsets are there? {b, c}, {b, d}, {c, d}
There are a total of 8 subsets of set A.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B
A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B A ⊂ C
A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B A ⊂ C A ⊂ B
C 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B A ⊂ C A ⊂ BC 6= A
C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A
B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A
∅ ⊂ A ∅ ⊂ C ∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A
∅ ⊂ C ∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C
∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: If A = {−3,−1, 1, 3}, B = {3,−3, 1,−1} andC = {−3,−2,−1, 0, 1, 2, 3}, each of the following statements istrue:
A = B A ⊂ C A ⊂ BC 6= A C 6⊂ A B ⊂ A∅ ⊂ A ∅ ⊂ C ∅ 6∈ A
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
DefinitionThe union of two sets A and B is the set of all elements formedby combining all the elements of set A and all the elements ofset B into one set. It is written A ∪ B. Symbolically,
A ∪ B = {x |x ∈ A or x ∈ B}
U
A B
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
DefinitionThe intersection of two sets A and B is the set of all elementsthat are common to both A and B. It is written A ∩ B.Symbolically,
A ∩ B = {x |x ∈ A and x ∈ B}
U
A B
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
If two sets have no elements in common, they are said tobe disjoint. Two sets A and B are disjoint if A ∩ B = ∅.
The set of all elements under consideration is called theuniversal set U. Once the universal set is determined for aparticular problem, all other sets under consideration mustbe subsets of U.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
If two sets have no elements in common, they are said tobe disjoint. Two sets A and B are disjoint if A ∩ B = ∅.The set of all elements under consideration is called theuniversal set U. Once the universal set is determined for aparticular problem, all other sets under consideration mustbe subsets of U.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
DefinitionThe complement of a set A is defined as the set of elementsthat are contained in U, the universal set, but not contained inset A. We donte the complement by A′. Symbolically,
A′ = {x ∈ U|x 6∈ A}
U
A B
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
Example: A marketing survey of 1,000 commuters found that600 answered listen to the news, 500 listen to music, and 300listen to both. Let N = set of commuters in the sample wholisten to news and M = set of commuters in the sample wholisten to music. Find the number of commuters in the setN ∩M ′.
The number of elements in a set A is denoted by n(A), so inthis case we are looking for n(N ∩M ′).
The study is based on 1000 commuters, so n(U) = 1000.
The number of elements in the four sections in the Venndiagram need to add up to 1000.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
Example: A marketing survey of 1,000 commuters found that600 answered listen to the news, 500 listen to music, and 300listen to both. Let N = set of commuters in the sample wholisten to news and M = set of commuters in the sample wholisten to music. Find the number of commuters in the setN ∩M ′.
The number of elements in a set A is denoted by n(A), so inthis case we are looking for n(N ∩M ′).
The study is based on 1000 commuters, so n(U) = 1000.
The number of elements in the four sections in the Venndiagram need to add up to 1000.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
Example: A marketing survey of 1,000 commuters found that600 answered listen to the news, 500 listen to music, and 300listen to both. Let N = set of commuters in the sample wholisten to news and M = set of commuters in the sample wholisten to music. Find the number of commuters in the setN ∩M ′.
The number of elements in a set A is denoted by n(A), so inthis case we are looking for n(N ∩M ′).
The study is based on 1000 commuters, so n(U) = 1000.
The number of elements in the four sections in the Venndiagram need to add up to 1000.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
Example: A marketing survey of 1,000 commuters found that600 answered listen to the news, 500 listen to music, and 300listen to both. Let N = set of commuters in the sample wholisten to news and M = set of commuters in the sample wholisten to music. Find the number of commuters in the setN ∩M ′.
The number of elements in a set A is denoted by n(A), so inthis case we are looking for n(N ∩M ′).
The study is based on 1000 commuters, so n(U) = 1000.
The number of elements in the four sections in the Venndiagram need to add up to 1000.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
U = 1, 000
M N
N ∩M ′
300200 300
So, 300 commuters listen to news and not music.
Math 1300 Finite Mathematics
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Set Properties and Set NotationSet Operations
U = 1, 000
M N
N ∩M ′300
200 300
So, 300 commuters listen to news and not music.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
U = 1, 000
M N
N ∩M ′300200
300
So, 300 commuters listen to news and not music.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
U = 1, 000
M N
300200 300
So, 300 commuters listen to news and not music.
Math 1300 Finite Mathematics
university-logo
Set Properties and Set NotationSet Operations
U = 1, 000
M N
300200 300
So, 300 commuters listen to news and not music.
Math 1300 Finite Mathematics