Math 1300: Section 7-2 Sets

Set Properties and Set Notation
Set Operations

Math 1300 Finite Mathematics
Section 7-2: Sets

Department of Mathematics
University of Missouri

October 16, 2009

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Set Operations

A set is any collection of objects speciﬁed in such a way
that we can determine whether a given object is or is not in
the collection.

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Set Operations

the collection.
Capital letters, such as A, B, and C are often used to
designate particular sets.

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Set Operations

the collection.
Capital letters, such as A, B, and C are often used to
designate particular sets.
Each object in a set is called a member or element of the
set.

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Set Operations

Symbolically,

a∈A means "a is an element of set A"
a∈A means "a is not an element of set A"

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Set Operations

Symbolically,


A set without any elements is called the empty, or null,
set.

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Set Operations

Symbolically,


A set without any elements is called the empty, or null,
set.
Symbollically

∅ denotes the empty set

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Set Operations

A set is usually described either by listing all its elements
between braces { } or by enclosing a rule within braces that
determines the elements of the set.

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Set Operations

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”

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Set Operations



Example:
Rule Listing
{x|x is a weekend day} = {Saturday, Sunday}
{x|x 2 = 4} = {−2, 2}
{x|x is a positive odd number} = {1, 3, 5, . . . }

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Set Operations



Example:
Rule Listing
{x|x is a weekend day} = {Saturday, Sunday}
{x|x 2 = 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 is
an infinite set.
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Set Operations

When listing the elements of a set, we do not list an
element more than once, and the order in which the
elements are listed does not matter.

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Set Operations

A is a subset of B if every element of A is also an element
of B.

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Set Operations

of B.
If A and B have exactly the same elments, then the two
sets are said to be equal.

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Set Operations

of B.
Symbollically,
A ⊂ B means “A is a subset of B”
A = B means “A equals B”
A ⊂ B means “A is not a subset of B”
A = B means “A and B do not have the same elements”

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Set Operations

of B.
Symbollically,
A ⊂ B means “A is a subset of B”
A = B means “A equals B”
A ⊂ B means “A is not a subset of B”
A = B means “A and B do not have the same elements”
∅ is a subset of every set.
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Set Operations

Example: List all subsets of the set A = {b, c, d}.

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Set Operations

First, ∅ ⊂ A.

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Set Operations

First, ∅ ⊂ A.
Also, every set is a subset of itself, so A ⊂ A.

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Set Operations

First, ∅ ⊂ A.
What one-element subsets?

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Set Operations

First, ∅ ⊂ A.
What one-element subsets? {b}, {c} and {d}.

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Set Operations

First, ∅ ⊂ A.
What two-element subsets are there?

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Set Operations

First, ∅ ⊂ A.
What two-element subsets are there? {b, c}, {b, d}, {c, d}

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Set Operations

First, ∅ ⊂ A.
What two-element subsets are there? {b, c}, {b, d}, {c, d}
There are a total of 8 subsets of set A.

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
C = {−3, −2, −1, 0, 1, 2, 3}, each of the following statements is
true:

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
true:
A=B

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
true:
A=B A⊂C

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
true:
A=B A⊂C A⊂B

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
true:
A=B A⊂C A⊂B
C=A

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
true:
A=B A⊂C A⊂B
C=A C⊂A

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
true:
A=B A⊂C A⊂B
C=A C⊂A B⊂A

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
true:
A=B A⊂C A⊂B
C=A C⊂A B⊂A
∅⊂A

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
true:
A=B A⊂C A⊂B
C=A C⊂A B⊂A
∅⊂A ∅⊂C

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Set Operations

Example: If A = {−3, −1, 1, 3}, B = {3, −3, 1, −1} and
true:
A=B A⊂C A⊂B
C=A C⊂A B⊂A
∅⊂A ∅⊂C ∅∈A

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Set Operations

Deﬁnition
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. Symbolically,

A ∪ B = {x|x ∈ A or x ∈ B}

U
A B

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Set Operations

Deﬁnition
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.
Symbolically,

A ∩ B = {x|x ∈ A and x ∈ B}

U
A B

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Set Operations

If two sets have no elements in common, they are said to
be disjoint. Two sets A and B are disjoint if A ∩ B = ∅.

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Set Operations

If two sets have no elements in common, they are said to
be disjoint. Two sets A and B are disjoint if A ∩ B = ∅.
The set of all elements under consideration is called the
universal set U. Once the universal set is determined for a
particular problem, all other sets under consideration must
be subsets of U.

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Set Operations

Deﬁnition
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. We donte the complement by A . Symbolically,

A = {x ∈ U|x ∈ A}

U
A B

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Set Operations

Example: A marketing survey of 1,000 commuters found that
600 answered 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 and M = set of commuters in the sample who
listen to music. Find the number of commuters in the set
N ∩M .

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Set Operations

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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Set Operations

N ∩M .
The study is based on 1000 commuters, so n(U) = 1000.

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Set Operations

N ∩M .
The study is based on 1000 commuters, so n(U) = 1000.
The number of elements in the four sections in the Venn
diagram need to add up to 1000.

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Set Operations

U = 1, 000

M N

N ∩M

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Set Operations

U = 1, 000

M N

300 N ∩M

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Set Operations

U = 1, 000

M N

200 300 N ∩M

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Set Operations

U = 1, 000

M N

200 300 300

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Set Operations

U = 1, 000

M N

200 300 300

So, 300 commuters listen to news and not music.

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Math 1300: Section 7-2 Sets

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