Sets
MATH 107: Finite Mathematics
University of Louisville
February 24, 2014
Set Basics
Describing Sets
2 / 18
What is a set?
A set is an
unordered
structure containing any number of
distinct
objects.
These objects (called
elements
of the set) can be anything: numbers,
words, people, functions even other sets!
Conventionally we label a set with a
capital letter,
and list its elements
in curly braces.
Several descriptions of the same set
▸
▸
▸
▸
A = {1, 3, 4}.
A is the set with elements
A = {4, 1, 3}.
A = {3, 1, 3, 4, 1, 1}.
MATH 107 (UofL)
1, 3, and 4.
Notes
February 24, 2014
Set Basics
Describing Sets
3 / 18
Describing sets by patterns
Often it is tedious to list out all the elements of a set, so we might use
ellipses to describe that many omitted elements follow a certain
pattern.
For instance, we might write
{1, 2, 3, . . . , 50}
instead of actually
writing out a set containing the rst 50 positive integers.
Any pattern of membership can be shortened using ellipsis once the
pattern is clear.
Many sets described by patterns
▸
▸
▸
▸
A = {−21, −16, −11, . . . , 44}.
B = {2, 3, 5, 7, 11, 13, 17, . . . , 97}.
C = {1, 2, 4, 8, 16, . . . , 1024}.
D = {Washington, Adams, Jeerson, . . . , Obama}.
MATH 107 (UofL)
Notes
Set Basics
February 24, 2014
Describing Sets
4 / 18
Innite sets
The pattern-following description becomes particularly useful when our
list would need
innitely
many entries. Compare the two following
descriptions:
A = {1, 3, 5, 7, . . . , 99}
and
B = {1, 3, 5, 7, . . .}
A constists only of the 50 odd positive numbers less than
while B contains every odd positive number. We can describe
Note that
100,
innite sets with patterns too:
Examples of innite sets
▸
▸
▸
▸
S = {1, 4, 9, 16, 25, . . .}.
T = { 12 , 13 , 14 , 15 , . . .}.
X = {3, 7, 11, 15, 19, . . .}.
Y = {. . . , −9, −5, −1, 3, 7, 11, 15, 19, . . .}.
MATH 107 (UofL)
Notes
February 24, 2014
Set Basics
Describing Sets
5 / 18
Describing sets by rules
Instead of using an ellipsis-described pattern, we can use a rule
instead, with a specic format:
S = {<variable>∣⟨condition⟩}
This template isn't obvious, so let's look at some examples:
How rules might describe a set
▸
{x ∣x
▸
{t ∣t 3 − t = 0} = {−1, 0, 1}.
▸
is a positive integer}
= {1, 2, 3, . . .}.
{k ∣k is a president of the U.S.} =
{Washington, Adams, Jeerson, . . . , Obama}.
MATH 107 (UofL)
Notes
Set Basics
February 24, 2014
Membership and special cases
6 / 18
Membership in sets
For any object
x
and set
A,
either
x
is an element of
A
or it is not.
We denote these two possibilities with the symbolic forms
x ∉A
x ∈A
respectively.
Examples of membership
▸ 2
∉ {1, 3, 4}.
▸ For
▸ 18
A = {1, 5, 6, 10},
5
∈ A.
∈ {2, 4, 6, 8, . . . , 100}.
▸ For
B = { 21 , 23 , 34 , 45 , . . .},
MATH 107 (UofL)
5
7
∉ B.
Notes
February 24, 2014
and
Set Basics
Membership and special cases
7 / 18
A special set or two
One set is a rather unusual one: a set can contain any number of
elements, including none at all.
Denition
The
empty set
or
null set,
denoted
∅
or
{},
is the set containing no
elements.
Another special set is somewhat context-dependent: the
denoted
U,
universal set,
is the set of all objects of interest. What objects are
interesting depends on the sets you're looking at:
Universal set examples
A = {−21, −16, −11, . . . , 44}, then U might contain all integers.
1 2 3 4
If B = { , , , , . . .}, then U might contain all rational numbers.
2 3 4 5
If C = {Washington, Adams, Jeerson, . . . , Obama}, then U might
▸ If
▸
▸
contain all people, living or dead.
MATH 107 (UofL)
Notes
February 24, 2014
Set operations
8 / 18
Introduction of set operations
As with logical statements, sets can be put together using some novel
operations. These operations will be as follows:
▸ The
▸ The
▸ The
union of two sets.
intersection of two sets.
complement of a set.
We will discuss each of these operations individually.
MATH 107 (UofL)
Notes
February 24, 2014
Set operations
9 / 18
Set unions
Denition
The
union
A and B , denoted A ∪ B ,
either A or B or both.
of sets
which are in
So, for instance, if
is the set of all objects
A = {1, 3, 5}, B = {1, 2, 4},
and
C = {2, 4, 6, 8, . . .}:
A ∪ B = {1, 2, 3, 4, 5}.
▸ A ∪ C = {1, 2, 3, 4, 5, 6, 8, 10, 12, . . .}.
▸ B ∪ C = {1, 2, 4, 6, 8, 10, 12, . . .}.
▸ A ∪ ∅ = {1, 3, 5}.
Note the symbolic similarity of A ∪ B (the set of things in A or B ) to
p ∨ q (the statement which is true when p or q is true). This is made
▸
explicit by a rule-based denition of the union:
Denition (Alternative denition of the union)
A ∪ B = {x ∣(x ∈ A) ∨ (x ∈ B )}
MATH 107 (UofL)
Notes
February 24, 2014
Set operations
10 / 18
Set intersections
Denition
The
intersection
of sets
A
objects which are in both
So, for instance, if
▸
▸
▸
▸
B , denoted A ∩ B ,
A and B .
and
A = {1, 3, 5}, B = {1, 2, 4},
A ∩ B = {1}.
A ∩ C = ∅.
B ∩ C = {2, 4}.
C ∩ ∅ = ∅.
Note the symbolic similarity of
B)
to
p∧q
A∩B
and
is the set of all
C = {2, 4, 6, 8, . . .}:
(the set of things in both
(the statement which is true when
p and q
are true). This
is made explicit by a rule-based denition of the intersection:
Denition (Alternative denition of the intersection)
A ∩ B = {x ∣(x ∈ A) ∧ (x ∈ B )}
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Notes
A and
February 24, 2014
Set operations
11 / 18
Set complements
Denition
complement
not in A.
The
are
of a set
A,
Note that we need a universal set
For instance, if
A′
denoted
U
is the set of all objects which
to describe all objects.
A = {1, 3, 5}, B = {1, 2, 4}, C = {2, 4, 6, . . .},
U = {1, 2, 3, . . .}:
▸ A′ = {2, 4, 6, 7, 8, 9, . . .}.
▸ B ′ = {3, 5, 6, 7, 8, 9, . . .}.
▸ C ′ = {1, 3, 5, 7, 9, 11, . . .}.
Note the symbolic similarity of
A′
(the statement which is true when
(the set of things
p
is
not
not
in
and
A)
to
¬p
true). This is made explicit
by a rule-based denition of the complement:
Denition (Alternative denition of the complement)
A′ = {x ∣¬(x ∈ A)}
MATH 107 (UofL)
Notes
February 24, 2014
Set tests
12 / 18
Tests on sets
There are a few important questions to be asked about sets:
▸ Is a given object in a set
▸ Does a particular set
A
A?
lie entirely within another set
B?
▸ Are two sets identical?
The rst we have already seen; the second and third we will describe
in greater detail.
MATH 107 (UofL)
Notes
February 24, 2014
Set tests
13 / 18
Subsets
Denition
A
of
is a
B.
subset
B,
of
For instance, if
denoted
A ⊂ B,
if every element of
A
is an element
A = {1, 3, 5}, B = {1, 2, 4}, C = {2, 4, 6, 8, . . .},
and
U = {1, 2, 3, . . .}, we might note some subset inclusions:
▸ ∅ ⊂ A, ∅ ⊂ B , and ∅ ⊂ C .
▸ A ⊂ U , B ⊂ U , and C ⊂ U .
▸ A⊂
/ C , B ⊂/ C , A ⊂/ B .
▸ A ⊂ C ′.
This concept is slightly connected to the idea of implication: the
statement A ⊂ B is true only when x ∈ A Ô⇒ x ∈ B is always true.
MATH 107 (UofL)
Notes
February 24, 2014
Set tests
14 / 18
Equality
Denition
A
is equal to
B,
denoted
A = B,
if
A
and
B
have the exact same
elements.
In particular, the statement that
both
A⊂B
and
MATH 107 (UofL)
B ⊂ A.
A=B
Notes
is the same as claiming that
February 24, 2014
Visualizing sets
15 / 18
Venn diagrams
To visualize a set and operations on it, the most common tool is a
Venn diagram,
which displays regions associated with every possible
set membership.
A
A
B
B
U
U
MATH 107 (UofL)
C
Notes
February 24, 2014
Visualizing sets
16 / 18
Set operations visualized with Venn diagrams
Using Venn diagrams, we can specically consider regions associated
with specic set operations.
A
A
B
U
U
(A ∪ B )′
A′ ∩ B ′
We may thus conclude, from this picture, that
MATH 107 (UofL)
B
Notes
A ′ ∩ B ′ = (A ∪ B ) ′ .
February 24, 2014
Visualizing sets
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Other rules governing set operations
All of these could be proven by drawing pictures, but we aren't going
to.
Double complement
Absorption of
∪
Absorption of
∩
Commutativity of
∪
Commutativity of
∩
Associativity of
∪
Associativity of
∩
DeMorgan's laws
Distributivity
(A′ )′ = A
B ⊂ A, then A ∪ B = A.
If B ⊂ A, then A ∩ B = B .
A ∪ B = B ∪ A.
A ∩ B = B ∩ A.
(A ∪ B ) ∪ C = A ∪ (B ∪ C )
(A ∩ B ) ∩ C = A ∩ (B ∩ C )
(A ′ ∪ B ′ ) = ( A ∩ B ) ′
(A ′ ∩ B ′ ) = ( A ∪ B ) ′
(A ∪ B ) ∩ C = ( A ∩ C ) ∪ ( B ∩ C )
( A ∩ B ) ∪ C = (A ∪ C ) ∩ ( B ∪ C )
If
MATH 107 (UofL)
Notes
February 24, 2014
Visualizing sets
18 / 18
Using Venn diagrams for counting purposes
We can use a Venn diagram to understand the
sizes
of sets.
A sample question
There are 70 students in this class. 19 of them have a grade of A, 22
are rst-years, and 8 rst-years have an A. How many students are
neither rst-year nor currently getting an A?
Let
U
be all the students,
A
those getting an A, and
A
8
A ∩ B. B
has 8 elements.
14
U
37
A∩B
A
has 19 elements, 8 of which are in
has 22 elements, 8 of which are in
elements, 33 of which are in
MATH 107 (UofL)
the rst-years.
B
11
We know
B
A ∪ B.
Notes
A ∩ B. U
has 70
February 24, 2014