CMSC 56 | Lecture 6: Sets & Set Operations

Sets & Set Operations
Lecture 6, CMSC 56
Allyn Joy D. Calcaben

Definition 1
A set is an unordered collection of objects, called elements or
members of the set. A set is said to contain its elements. We
write a ∈ A to denote that a is an element of the set A. The
notation a ∈ A denotes that a is not an element of the set A.
Set

Definition 1
A set is an unordered collection of objects, called elements or
members of the set. A set is said to contain its elements. We
write a ∈ A to denote that a is an element of the set A. The
notation a ∈ A denotes that a is not an element of the set A.
Number of objects in a set can be finite or infinite.
Set

a set of chairs
the set of nobel laureates in the world
Example

a set of chairs
the set of integers
Example

a set of chairs
the set of integers
the set of natural numbers less than 10
Example

a set of chairs
the set of integers
the set of natural numbers less than 10
the set of students in this class who passed the quizzes
Example

One way of describing a set.
All members of the set are listed between curly braces, and
separated by comma.
Roster Method

The set V of all vowels in the English alphabet can be written
as V = {a, e, i, o, u}.
Example

The set V of all vowels in the English alphabet can be written
as V = {a, e, i, o, u}.
The set O of odd positive integers less than 10 can be
expressed by O = {1,3,5,7,9}.
Example

Another way to describe a set.
Characterized all those elements in the set by stating the
property or properties they must have to be members.
Set Builder

The set O of all odd positive integers less than 10.
Example

O = { x | x is an odd positive integer less than 10 }
Example

O = { x | x is an odd positive integer less than 10 }
or,
O = { x ∈ Z+ | x is odd and x < 10 }
Example

N = { 0, 1, 2, 3,... }, the set of natural numbers
Z = { …, -2, -1, 0, 1, 2, … }, the set of integers
Z+ = { 1, 2, 3, … }, the set of positive integers
Q = { p/q | p ∈ Z, q ∈ Z, and q ≠ 0 }, the set of rational numbers
R the set of real numbers
R+ the set of positive real numbers
C the set of complex numbers
Important roles

When a and b are real numbers with a < b, we write
[ a , b ] = { x | a ≤ x ≤ b }
[ a , b ) = { x | a ≤ x < b }
( a , b ] = { x | a < x ≤ b }
( a , b ) = { x | a < x < b }
Intervals of real numbers

When a and b are real numbers with a < b, we write
[ a , b ] = { x | a ≤ x ≤ b }
[ a , b ) = { x | a ≤ x < b }
( a , b ] = { x | a < x ≤ b }
( a , b ) = { x | a < x < b }
Note that: [ a , b ] is called the closed interval from a to b.
( a , b ) is called the open interval from a to b.
Intervals of real numbers

Definition 2
Two sets are equal if and only if they have the same elements.
Therefore, if A and B are sets, then A and B are equal if and
only if ∀x ( x ∈ A ↔ x ∈ B ). We write A = B if A and B are
equal sets.
Equality

{ 1, 3, 5 } = { 3, 5, 1 }
Example

{ 1, 3, 5 } = { 3, 5, 1 } = { 1, 3, 3, 3, 5, 5, 5, 5, 5 }
Example

{ 1, 3, 5 } = { 3, 5, 1 } = { 1, 3, 3, 3, 5, 5, 5, 5, 5 }
Note:
It doesn’t matter if an element of a set is listed more
than once as long as they have the same elements.
Example

The universal set is denoted by U: the set of all objects under
the consideration, represented by a rectangle.
The empty set is denoted as Ø or { }.
Special Sets

U
Example
A Venn Diagram that
represents V which has
the set of vowels in the
English alphabet.

U
Example
A Venn Diagram that
represents V which has
the set of vowels in the
English alphabet.
V = { a, b, c, d, e }
a
e
i
o
u
VV

Definition 3
The set A is a subset of B if and only if every element of A is
also an element of B. We use the notation A ⊆ B to indicate
that A is a subset of the set B.
Subset

Subset
U
B
A
Alternate way to
define A is a subset of B is
∀x ( x ∈ A → x ∈ B )

Theorem
For every set S,
(i) Ø ⊆ S
(ii) S ⊆ S.
Empty set / Subset Properties

Proof
Recall the definition of a subset:
All elements of a set A must be also elements of B:
∀x ( x ∈ A → x ∈ B )

Proof
∀x ( x ∈ A → x ∈ B )
We must show the following implication holds for any S
∀x ( x ∈ Ø → x ∈ S )

Proof
∀x ( x ∈ A → x ∈ B )
∀x ( x ∈ Ø → x ∈ S )
Since the empty set does not contain any element,
x ∈ Ø is always FALSE.

Proof
∀x ( x ∈ A → x ∈ B )
∀x ( x ∈ Ø → x ∈ S )
Since the empty set does not contain any element,
x ∈ Ø is always FALSE. Then the implication is always TRUE.

Definition 3.5
A set A is said to be a proper subset of B if and only if A ⊆ B
and A ≠ B. We denote that A is a proper subset of B with the
notation A ⊂ B
Proper Subset

Example
U
B
A
A = { 1, 2, 3 }
B = { 1, 2, 3, 4, 5, 6 }
Is A ⊂ B?12
3
4
5
6

Example
U
B
A
A = { 1, 2, 3 }
B = { 1, 2, 3, 4, 5, 6 }
Is A ⊂ B? YES
12
3
4
5
6

Definition 4
Let S be a set. It there are exactly n distinct elements in S
where n is a nonnegative integer, we say that S is a finite set
and that n is the cardinality of S. The cardinality of S is
denoted by |S|.
Cardinality

V = { 1, 2, 3, 4, 5 }
| V | = 5
Example

V = { 1, 2, 3, 4, 5 }
| V | = 5
A = { 1, 2, 3, 4, 5, … , 20 }
| A | = 20
Example

V = { 1, 2, 3, 4, 5 }
| V | = 5
A = { 1, 2, 3, 4, 5, … , 20 }
| A | = 20
| Ø | = 0
Example

Definition 5
A set is infinite if it is not finite.
Infinite Set

The set of natural numbers is an infinite set.
N = { 1, 2, 3, … }
The set of reals is an infinite set.
Example

Definition 6
Given a set S, the power set of S is the set of all subsets of the
set S. The power set of S is denoted by P(S).
Power Set

What is the power set of the set { 0, 1, 2 }?
Example

What is the power set of the set { 0, 1, 2 }?
The power set P({ 0, 1, 2 }) is the set of all subsets of { 0, 1, 2 }.
Hence,
P({ 0, 1, 2 }) = { Ø, {0}, {1}, {2}, { 0, 1 }, { 0, 2 }, { 1, 2 }, { 0, 1, 2 }}
Solution

Definition 7
The ordered n-tuple ( a1 ,a2 , … , an ) is the ordered collection
that has a1 as its first element, a2 as its second element, …,
and an as its nth element.
n-tuple

Definition 8
Let S and T be sets. The Cartesian Product of A and B, denoted
by A x B, is the set of all ordered pairs (a, b), where a ∈ A and
b ∈ B. Hence,
A x B = { (a, b) | a ∈ A ꓥ b ∈ B }
Cartesian Product

What is the Cartesian Product of A = { 2, 5 } and B = { x, y, z }?
Example

A x B = { (2, x), (2, y), (2, z), (5, x), (5, y), (5, z) }
Solution

A x B = { (2, x), (2, y), (2, z), (5, x), (5, y), (5, z) }
B x A = { (x, 2), (y, 2), (z, 2), (x, 5), (y, 5), (z, 5) }
Solution

A x B = { (2, x), (2, y), (2, z), (5, x), (5, y), (5, z) }
B x A = { (x, 2), (y, 2), (z, 2), (x, 5), (y, 5), (z, 5) }
Note:
S x T ≠ T x S
Solution

Definition 9
The Cartesian product of the sets A1, A2, …. , An, denoted by
A1 x A2 x …. x An, is the set of ordered n-tuples ( a1 ,a2 , … , an ),
where ai belongs to Ai for i = 1, 2, …, n. In other words,
A1 x A2 x …. x An = {( a1 ,a2 , … , an ) | ai ∈ Ai for i = 1, 2, …, n }
Cardinality of the Cartesian Product

A = { John, Peter, Mike }
B = { Jane, Ann, Laura }
A x B =
Example

A x B = { (John, Jane),(John, Ann) , (John, Laura), (Peter, Jane),
(Peter, Ann) , (Peter, Laura) , (Mike, Jane) , (Mike,
Ann) , (Mike, Laura) }
Solution

A x B = { (John, Jane),(John, Ann) , (John, Laura), (Peter, Jane),
(Peter, Ann) , (Peter, Laura) , (Mike, Jane) , (Mike,
Ann) , (Mike, Laura) }
| A x B | = 9
| A | = 3, | B | = 3
| A | * | B | = 9
Solution

Definition 1
Let A and B be sets. The union of the sets A and B, denoted by
A U B, is the set that contains those elements that are either
in A or B.
Union

U
Example
A = { a, b, c, d, o }
B = { a, e, i, o, u }
a
e
i
o
u
A
B
b
c
d
f
g

U
Solution
A = { a, b, c, d, o }
B = { a, e, i, o, u }
A U B = { a, b, c, d, e, i, o, u}
a
e
i
o
u
A
B
b
c
d
f
g

Definition 2
Let A and B be sets. The intersection of the sets A and B,
denoted by A ∩ B, is the set that contains those elements in
both A and B.
Intersection

U
Intersection
Alternate:
A ∩ B = { x | ( x ∈ A ꓥ x ∈ B }
a
e
i
o
u
A
B
b
c
d
f
g

U
Solution
A = { a, b, c, d, o }
B = { a, e, i, o, u }
A ∩ B = { a, o }
a
e
i
o
u
A
B
b
c
d
f
g

Definition 3
Two sets are called disjoint if their intersection is the empty
set.
Disjoint Sets

U
Disjoint
Alternate:
A and B are disjoint if and
only if A ∩ B = Ø
a
e
i
o
u
A
B
b
c
d
f
g

U
Example
A = { b, c, d }
B = { a, e, o }
A ∩ B =
a
e
i
o
u
A
B
b
c
d
f
g

U
Example
A = { b, c, d }
B = { a, e, o }
A ∩ B = { Ø }
DISJOINT!a
e
i
o
u
A
B
b
c
d
f
g

Definition 4
Let A and B be sets. The difference of the sets A and B,
denoted by A – B, is the set that contains those elements that
are in A but not in B. The difference of A and B is also called
the compliment of B with respect to A.
Set Difference

Definition 4
Let A and B be sets. The difference of the sets A and B,
denoted by A – B, is the set that contains those elements that
are in A but not in B. The difference of A and B is also called
the compliment of B with respect to A.
Note: It is sometimes denoted by AB
Set Difference

U
Set Difference
Alternate:
A – B = { x | ( x ∈ A ꓥ x ∈ B }
a
e
i
o
u
A
B
b
c
d
f
g

U
Example
A = { a, b, c, d, o }
B = { a, e, i, o, u }
A – B =
a
e
i
o
u
A
B
b
c
d
f
g

U
Example
A = { a, b, c, d, o }
B = { a, e, i, o, u }
A – B = { b, c, d }
a
e
i
o
u
A
B
b
c
d
f
g

Definition 5
Let U be the universal set. The compliment of the set A,
denoted by Ā, is the compliment of A with respect to U.
Therefore, the compliment of the set A is U – A.
Compliment

Let A = { a, e, i, o, u }
where the universal set is the set of letters of the English
Alphabet.
Example

Let A = { a, e, i, o, u }
where the universal set is the set of letters of the English
Alphabet.
Ā = { b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z }
Solution

CMSC 56 | Lecture 6: Sets & Set Operations

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