Discrete Mathematics 1
Chapter 2: Basic Structures: Sets, Functions, Sequences and Sums
Department of Mathematics
The FPT university
TrungDT (FUHN)
MAD111 Chapter 2
1/1
2.1 Sets
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
A set is an unordered collection of objects.
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
A set is an unordered collection of objects.
An object of a set is called an element, or a member, of that set.
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
A set is an unordered collection of objects.
An object of a set is called an element, or a member, of that set.
The cardinality of the set A is the number of distinct elements of A,
denoted by |A|.
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
A set is an unordered collection of objects.
An object of a set is called an element, or a member, of that set.
The cardinality of the set A is the number of distinct elements of A,
denoted by |A|.
The empty set, denoted by Ø, is the set whose cardinality is 0.
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
A set is an unordered collection of objects.
An object of a set is called an element, or a member, of that set.
The cardinality of the set A is the number of distinct elements of A,
denoted by |A|.
The empty set, denoted by Ø, is the set whose cardinality is 0.
Example.
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
A set is an unordered collection of objects.
An object of a set is called an element, or a member, of that set.
The cardinality of the set A is the number of distinct elements of A,
denoted by |A|.
The empty set, denoted by Ø, is the set whose cardinality is 0.
Example.
The set {a, cat, catches, a, mouse} has
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
A set is an unordered collection of objects.
An object of a set is called an element, or a member, of that set.
The cardinality of the set A is the number of distinct elements of A,
denoted by |A|.
The empty set, denoted by Ø, is the set whose cardinality is 0.
Example.
The set {a, cat, catches, a, mouse} has 4 elements.
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
A set is an unordered collection of objects.
An object of a set is called an element, or a member, of that set.
The cardinality of the set A is the number of distinct elements of A,
denoted by |A|.
The empty set, denoted by Ø, is the set whose cardinality is 0.
Example.
The set {a, cat, catches, a, mouse} has 4 elements.
The set {a, b, {a, b}, c, {a, b, c, },Ø} has
TrungDT (FUHN)
MAD111 Chapter 2
2/1
2.1 Sets
Definition
A set is an unordered collection of objects.
An object of a set is called an element, or a member, of that set.
The cardinality of the set A is the number of distinct elements of A,
denoted by |A|.
The empty set, denoted by Ø, is the set whose cardinality is 0.
Example.
The set {a, cat, catches, a, mouse} has 4 elements.
The set {a, b, {a, b}, c, {a, b, c, },Ø} has 6 elements
TrungDT (FUHN)
MAD111 Chapter 2
2/1
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A.
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x}
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
x ⊆ {x}
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
x ⊆ {x} (F)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
x ⊆ {x} (F)
{a, b} ⊆ {a, b, {a, b}, c}
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
x ⊆ {x} (F)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b} ∈ {a, b, {a, b}, c}
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
x ⊆ {x} (F)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b} ∈ {a, b, {a, b}, c} (T)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
Ø ⊆ {Ø}
x ⊆ {x} (F)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b} ∈ {a, b, {a, b}, c} (T)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
Ø ⊆ {Ø} (T)
x ⊆ {x} (F)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b} ∈ {a, b, {a, b}, c} (T)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
Ø ⊆ {Ø} (T)
x ⊆ {x} (F)
Ø ∈ {Ø}
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b} ∈ {a, b, {a, b}, c} (T)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
Ø ⊆ {Ø} (T)
x ⊆ {x} (F)
Ø ∈ {Ø} (T)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b} ∈ {a, b, {a, b}, c} (T)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
Ø ⊆ {Ø} (T)
x ⊆ {x} (F)
Ø ∈ {Ø} (T)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b, c} ⊆ {a, b, c}
{a, b} ∈ {a, b, {a, b}, c} (T)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
Ø ⊆ {Ø} (T)
x ⊆ {x} (F)
Ø ∈ {Ø} (T)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b, c} ⊆ {a, b, c} (T)
{a, b} ∈ {a, b, {a, b}, c} (T)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
Ø ⊆ {Ø} (T)
x ⊆ {x} (F)
Ø ∈ {Ø} (T)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b, c} ⊆ {a, b, c} (T)
{a, b} ∈ {a, b, {a, b}, c} (T)
{a, b, c} ∈ {a, b, c}
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
Ø ⊆ {Ø} (T)
x ⊆ {x} (F)
Ø ∈ {Ø} (T)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b, c} ⊆ {a, b, c} (T)
{a, b} ∈ {a, b, {a, b}, c} (T)
{a, b, c} ∈ {a, b, c} (F)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
If x is an element of A we write x ∈ A. If x is not an element of A we
write x 6∈ A.
If all elements of A are also elements of B we write A ⊆ B, and A is
called a subset of B.
If A is a proper subset of B, meaning A ⊆ B and A 6= B, we write
A ⊂ B.
The empty set Ø is a subset of any set.
Example. Which statements are true?
x ∈ {x} (T)
Ø ⊆ {Ø} (T)
x ⊆ {x} (F)
Ø ∈ {Ø} (T)
{a, b} ⊆ {a, b, {a, b}, c} (T)
{a, b, c} ⊆ {a, b, c} (T)
{a, b} ∈ {a, b, {a, b}, c} (T)
{a, b, c} ∈ {a, b, c} (F)
TrungDT (FUHN)
MAD111 Chapter 2
3/1
TrungDT (FUHN)
MAD111 Chapter 2
4/1
Definitions
TrungDT (FUHN)
MAD111 Chapter 2
4/1
Definitions
The Cartesian product of two sets A and B, denoted by A × B, is the
set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
TrungDT (FUHN)
MAD111 Chapter 2
4/1
Definitions
The Cartesian product of two sets A and B, denoted by A × B, is the
set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
The power set of the set A, denoted by P(A), is the set of all subsets
of A.
TrungDT (FUHN)
MAD111 Chapter 2
4/1
Definitions
The Cartesian product of two sets A and B, denoted by A × B, is the
set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
The power set of the set A, denoted by P(A), is the set of all subsets
of A.
Note.
TrungDT (FUHN)
MAD111 Chapter 2
4/1
Definitions
The Cartesian product of two sets A and B, denoted by A × B, is the
set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
The power set of the set A, denoted by P(A), is the set of all subsets
of A.
Note.
If |A| = m and |B| = n then |A × B| = mn.
TrungDT (FUHN)
MAD111 Chapter 2
4/1
Definitions
The Cartesian product of two sets A and B, denoted by A × B, is the
set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
The power set of the set A, denoted by P(A), is the set of all subsets
of A.
Note.
If |A| = m and |B| = n then |A × B| = mn.
If |A| = n then |P(A)| = 2n .
TrungDT (FUHN)
MAD111 Chapter 2
4/1
2.2 Set Operations
TrungDT (FUHN)
MAD111 Chapter 2
5/1
2.2 Set Operations
Let A and B be two sets.
TrungDT (FUHN)
MAD111 Chapter 2
5/1
2.2 Set Operations
Let A and B be two sets.
Union of A and B:
TrungDT (FUHN)
A ∪ B = {x| (x ∈ A) ∨ (x ∈ B)}
MAD111 Chapter 2
5/1
2.2 Set Operations
Let A and B be two sets.
Union of A and B:
A ∪ B = {x| (x ∈ A) ∨ (x ∈ B)}
Intersection of A and B:
TrungDT (FUHN)
A ∩ B = {x| (x ∈ A) ∧ (x ∈ B)}
MAD111 Chapter 2
5/1
2.2 Set Operations
Let A and B be two sets.
Union of A and B:
A ∪ B = {x| (x ∈ A) ∨ (x ∈ B)}
Intersection of A and B:
Difference of A and B:
TrungDT (FUHN)
A ∩ B = {x| (x ∈ A) ∧ (x ∈ B)}
A − B = {x| (x ∈ A) ∧ (x 6∈ B)}
MAD111 Chapter 2
5/1
2.2 Set Operations
Let A and B be two sets.
Union of A and B:
A ∪ B = {x| (x ∈ A) ∨ (x ∈ B)}
Intersection of A and B:
Difference of A and B:
A ∩ B = {x| (x ∈ A) ∧ (x ∈ B)}
A − B = {x| (x ∈ A) ∧ (x 6∈ B)}
Symmetric difference of A and B:
TrungDT (FUHN)
A ⊕ B = {x| (x ∈ A) ⊕ (x ∈ B)}
MAD111 Chapter 2
5/1
2.2 Set Operations
Let A and B be two sets.
Union of A and B:
A ∪ B = {x| (x ∈ A) ∨ (x ∈ B)}
Intersection of A and B:
Difference of A and B:
A ∩ B = {x| (x ∈ A) ∧ (x ∈ B)}
A − B = {x| (x ∈ A) ∧ (x 6∈ B)}
Symmetric difference of A and B:
A ⊕ B = {x| (x ∈ A) ⊕ (x ∈ B)}
Complement of A with respect to the universal set U: A = U − A
TrungDT (FUHN)
MAD111 Chapter 2
5/1
Set Identities
TrungDT (FUHN)
MAD111 Chapter 2
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Set Identities
Name
TrungDT (FUHN)
Identity
MAD111 Chapter 2
6/1
Set Identities
Name
Identity
Complementation law
(A) = A
TrungDT (FUHN)
MAD111 Chapter 2
6/1
Set Identities
Name
Identity
Complementation law
Identity laws
(A) = A
A ∪ Ø = A, A ∩ U = A
TrungDT (FUHN)
MAD111 Chapter 2
6/1
Set Identities
Name
Identity
Complementation law
Identity laws
Domination laws
(A) = A
A ∪ Ø = A, A ∩ U = A
A ∪ U = U, A ∩ Ø = Ø
TrungDT (FUHN)
MAD111 Chapter 2
6/1
Set Identities
Name
Identity
Complementation law
Identity laws
Domination laws
Complement laws
(A) = A
A ∪ Ø = A, A ∩ U = A
A ∪ U = U, A ∩ Ø = Ø
A ∪ A = U, A ∩ A = Ø
TrungDT (FUHN)
MAD111 Chapter 2
6/1
Set Identities
Name
Identity
Complementation law
Identity laws
Domination laws
Complement laws
Idempotent laws
(A) = A
A ∪ Ø = A,
A ∪ U = U,
A ∪ A = U,
A ∪ A = A,
TrungDT (FUHN)
MAD111 Chapter 2
A∩U =A
A∩Ø=Ø
A∩A=Ø
A∩A=A
6/1
Set Identities
Name
Identity
Complementation law
Identity laws
Domination laws
Complement laws
Idempotent laws
Commutative laws
(A) = A
A ∪ Ø = A, A ∩ U = A
A ∪ U = U, A ∩ Ø = Ø
A ∪ A = U, A ∩ A = Ø
A ∪ A = A, A ∩ A = A
A ∪ B = B ∪ A, A ∩ B = B ∩ A
TrungDT (FUHN)
MAD111 Chapter 2
6/1
Set Identities
Name
Identity
Complementation law
Identity laws
Domination laws
Complement laws
Idempotent laws
Commutative laws
Associative laws
(A) = A
A ∪ Ø = A, A ∩ U = A
A ∪ U = U, A ∩ Ø = Ø
A ∪ A = U, A ∩ A = Ø
A ∪ A = A, A ∩ A = A
A ∪ B = B ∪ A, A ∩ B = B ∩ A
(A ∪ B) ∪ C = A ∪ (B ∪ C )
(A ∩ B) ∩ C = A ∩ (B ∩ C )
TrungDT (FUHN)
MAD111 Chapter 2
6/1
Set Identities
Name
Identity
Complementation law
Identity laws
Domination laws
Complement laws
Idempotent laws
Commutative laws
Associative laws
(A) = A
A ∪ Ø = A, A ∩ U = A
A ∪ U = U, A ∩ Ø = Ø
A ∪ A = U, A ∩ A = Ø
A ∪ A = A, A ∩ A = A
A ∪ B = B ∪ A, A ∩ B = B ∩ A
(A ∪ B) ∪ C = A ∪ (B ∪ C )
(A ∩ B) ∩ C = A ∩ (B ∩ C )
A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )
Distributive laws
TrungDT (FUHN)
MAD111 Chapter 2
6/1
Set Identities
Name
Identity
Complementation law
Identity laws
Domination laws
Complement laws
Idempotent laws
Commutative laws
Associative laws
(A) = A
A ∪ Ø = A, A ∩ U = A
A ∪ U = U, A ∩ Ø = Ø
A ∪ A = U, A ∩ A = Ø
A ∪ A = A, A ∩ A = A
A ∪ B = B ∪ A, A ∩ B = B ∩ A
(A ∪ B) ∪ C = A ∪ (B ∪ C )
(A ∩ B) ∩ C = A ∩ (B ∩ C )
A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )
Distributive laws
De Morgan’s laws
TrungDT (FUHN)
A∪B =A∩B
A∩B =A∪B
MAD111 Chapter 2
6/1
Two methods to show the equality of two sets.
TrungDT (FUHN)
MAD111 Chapter 2
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Two methods to show the equality of two sets.
Show that each set is a subset of the other.
TrungDT (FUHN)
MAD111 Chapter 2
7/1
Two methods to show the equality of two sets.
Show that each set is a subset of the other.
Use Membership table, similar to the method of using truth table to
establish propositional equivalences.
TrungDT (FUHN)
MAD111 Chapter 2
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Computer Representation of Sets
TrungDT (FUHN)
MAD111 Chapter 2
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Computer Representation of Sets
Let U be a universal set. Fix an ordering of elements of U as a1 , a2 , . . . , an .
TrungDT (FUHN)
MAD111 Chapter 2
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Computer Representation of Sets
Let U be a universal set. Fix an ordering of elements of U as a1 , a2 , . . . , an .
If A is a subset of U, represent A with a bit string of length n, where the
ith bit is 1 if ai is in A, and is 0 if ai is not in A.
TrungDT (FUHN)
MAD111 Chapter 2
8/1
Computer Representation of Sets
Let U be a universal set. Fix an ordering of elements of U as a1 , a2 , . . . , an .
If A is a subset of U, represent A with a bit string of length n, where the
ith bit is 1 if ai is in A, and is 0 if ai is not in A.
Example.
TrungDT (FUHN)
MAD111 Chapter 2
8/1
Computer Representation of Sets
Let U be a universal set. Fix an ordering of elements of U as a1 , a2 , . . . , an .
If A is a subset of U, represent A with a bit string of length n, where the
ith bit is 1 if ai is in A, and is 0 if ai is not in A.
Example. Let U = {1, 2, 3, 4, 5, 6, 7, 8}.
TrungDT (FUHN)
MAD111 Chapter 2
8/1
Computer Representation of Sets
Let U be a universal set. Fix an ordering of elements of U as a1 , a2 , . . . , an .
If A is a subset of U, represent A with a bit string of length n, where the
ith bit is 1 if ai is in A, and is 0 if ai is not in A.
Example. Let U = {1, 2, 3, 4, 5, 6, 7, 8}. Then the subset A = {1, 3, 4, 6}
is represented as the bit string 10110100.
TrungDT (FUHN)
MAD111 Chapter 2
8/1
2.3 Functions
TrungDT (FUHN)
MAD111 Chapter 2
9/1
2.3 Functions
TrungDT (FUHN)
MAD111 Chapter 2
9/1
2.3 Functions
TrungDT (FUHN)
MAD111 Chapter 2
9/1
2.3 Functions
TrungDT (FUHN)
MAD111 Chapter 2
9/1
2.3 Functions
The set A is called domain
and B is called codomain of f .
TrungDT (FUHN)
MAD111 Chapter 2
9/1
2.3 Functions
The set A is called domain
and B is called codomain of f .
If f (a) = b we say b is the image of a
and a is a preimage of b.
TrungDT (FUHN)
MAD111 Chapter 2
9/1
2.3 Functions
The set A is called domain
and B is called codomain of f .
If f (a) = b we say b is the image of a
and a is a preimage of b.
Let S be a subset of A. The set
f (S) = {b ∈ B|∃a ∈ A(f (a) = b)}
is called the image of S,
TrungDT (FUHN)
MAD111 Chapter 2
9/1
2.3 Functions
The set A is called domain
and B is called codomain of f .
If f (a) = b we say b is the image of a
and a is a preimage of b.
Let S be a subset of A. The set
f (S) = {b ∈ B|∃a ∈ A(f (a) = b)}
is called the image of S, and the set
f −1 (S) = {a ∈ A|f (a) ∈ S}
is called the preimage of S.
TrungDT (FUHN)
MAD111 Chapter 2
9/1
2.3 Functions
The set A is called domain
and B is called codomain of f .
If f (a) = b we say b is the image of a
and a is a preimage of b.
Let S be a subset of A. The set
f (S) = {b ∈ B|∃a ∈ A(f (a) = b)}
is called the image of S, and the set
f −1 (S) = {a ∈ A|f (a) ∈ S}
is called the preimage of S. The set f (A) is called the range of f .
TrungDT (FUHN)
MAD111 Chapter 2
9/1
Some Important Functions
TrungDT (FUHN)
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc
TrungDT (FUHN)
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
TrungDT (FUHN)
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
Ceiling function: dxe
TrungDT (FUHN)
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
Ceiling function: dxe = the smallest integer that is not smaller than x.
TrungDT (FUHN)
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
Ceiling function: dxe = the smallest integer that is not smaller than x.
Note. For all real numbers x we have
x − 1 < bxc ≤ x ≤ dxe < x + 1
TrungDT (FUHN)
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
Ceiling function: dxe = the smallest integer that is not smaller than x.
Note. For all real numbers x we have
x − 1 < bxc ≤ x ≤ dxe < x + 1
Example. Which statements are true for all real numbers x, y and all
integers n?
TrungDT (FUHN)
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
Ceiling function: dxe = the smallest integer that is not smaller than x.
Note. For all real numbers x we have
x − 1 < bxc ≤ x ≤ dxe < x + 1
Example. Which statements are true for all real numbers x, y and all
integers n?
dx + y e = dxe + dy e
TrungDT (FUHN)
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
Ceiling function: dxe = the smallest integer that is not smaller than x.
Note. For all real numbers x we have
x − 1 < bxc ≤ x ≤ dxe < x + 1
Example. Which statements are true for all real numbers x, y and all
integers n?
dx + y e = dxe + dy e
bx + y c = bxc + by c
TrungDT (FUHN)
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
Ceiling function: dxe = the smallest integer that is not smaller than x.
Note. For all real numbers x we have
x − 1 < bxc ≤ x ≤ dxe < x + 1
Example. Which statements are true for all real numbers x, y and all
integers n?
dx + y e = dxe + dy e
bx + y c = bxc + by c
TrungDT (FUHN)
dx + ne = dxe + n
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
Ceiling function: dxe = the smallest integer that is not smaller than x.
Note. For all real numbers x we have
x − 1 < bxc ≤ x ≤ dxe < x + 1
Example. Which statements are true for all real numbers x, y and all
integers n?
dx + y e = dxe + dy e
bx + y c = bxc + by c
TrungDT (FUHN)
dx + ne = dxe + n
bx + nc = bxc + n
MAD111 Chapter 2
10 / 1
Some Important Functions
Floor function: bxc = the greatest integer that is not greater than x.
Ceiling function: dxe = the smallest integer that is not smaller than x.
Note. For all real numbers x we have
x − 1 < bxc ≤ x ≤ dxe < x + 1
Example. Which statements are true for all real numbers x, y and all
integers n?
dx + y e = dxe + dy e
bx + y c = bxc + by c
TrungDT (FUHN)
dx + ne = dxe + n
bx + nc = bxc + n
MAD111 Chapter 2
10 / 1
One-to-One, Onto, and Bijection
TrungDT (FUHN)
MAD111 Chapter 2
11 / 1
One-to-One, Onto, and Bijection
The function f : A → B is one-to-one if f (a1 ) 6= f (a2 ) for all a1 6= a2 in A.
TrungDT (FUHN)
MAD111 Chapter 2
11 / 1
One-to-One, Onto, and Bijection
The function f : A → B is one-to-one if f (a1 ) 6= f (a2 ) for all a1 6= a2 in A.
TrungDT (FUHN)
MAD111 Chapter 2
11 / 1
TrungDT (FUHN)
MAD111 Chapter 2
12 / 1
Example. Which functions are one-to-one?
TrungDT (FUHN)
MAD111 Chapter 2
12 / 1
Example. Which functions are one-to-one?
(a) f : R → R;
TrungDT (FUHN)
f (x) = x 2
MAD111 Chapter 2
12 / 1
Example. Which functions are one-to-one?
(a) f : R → R;
(b) f : R+ → R;
TrungDT (FUHN)
f (x) = x 2
f (x) = x 2
MAD111 Chapter 2
12 / 1
Example. Which functions are one-to-one?
(a) f : R → R;
f (x) = x 2
(b) f : R+ → R;
f (x) = x 2
n+1
f (n) = b
c
2
(c) f : Z → Z;
TrungDT (FUHN)
MAD111 Chapter 2
12 / 1
Example. Which functions are one-to-one?
(a) f : R → R;
f (x) = x 2
(b) f : R+ → R;
f (x) = x 2
n+1
(c) f : Z → Z; f (n) = b
c
2
(d) f : Z × Z → Z; f (m, n) = m + n
TrungDT (FUHN)
MAD111 Chapter 2
12 / 1
TrungDT (FUHN)
MAD111 Chapter 2
13 / 1
The function f : A → B is onto if for each b in B there is a in A such that
f (a) = b.
TrungDT (FUHN)
MAD111 Chapter 2
13 / 1
The function f : A → B is onto if for each b in B there is a in A such that
f (a) = b. In other words, the function f : A → B is onto if f (A) = B.
TrungDT (FUHN)
MAD111 Chapter 2
13 / 1
The function f : A → B is onto if for each b in B there is a in A such that
f (a) = b. In other words, the function f : A → B is onto if f (A) = B.
TrungDT (FUHN)
MAD111 Chapter 2
13 / 1
TrungDT (FUHN)
MAD111 Chapter 2
14 / 1
Example. Which functions are onto:
TrungDT (FUHN)
MAD111 Chapter 2
14 / 1
Example. Which functions are onto:
(a) f : R → R;
TrungDT (FUHN)
f (x) = x 2
MAD111 Chapter 2
14 / 1
Example. Which functions are onto:
(a) f : R → R;
f (x) = x 2
(b) f : R → R;
f (x) = x 3
TrungDT (FUHN)
MAD111 Chapter 2
14 / 1
Example. Which functions are onto:
(a) f : R → R;
f (x) = x 2
(b) f : R → R;
f (x) = x 3
(c) f : R → Z;
f (x) = 2bxc
TrungDT (FUHN)
MAD111 Chapter 2
14 / 1
Example. Which functions are onto:
(a) f : R → R;
f (x) = x 2
(b) f : R → R;
f (x) = x 3
(c) f : R → Z;
f (x) = 2bxc
(d) f : R → Z;
f (x) = b2xc
TrungDT (FUHN)
MAD111 Chapter 2
14 / 1
Example. Which functions are onto:
(a) f : R → R;
f (x) = x 2
(b) f : R → R;
f (x) = x 3
(c) f : R → Z;
f (x) = 2bxc
(d) f : R → Z;
f (x) = b2xc
(e) f : Z × Z → Z;
TrungDT (FUHN)
f (m, n) = m + n
MAD111 Chapter 2
14 / 1
TrungDT (FUHN)
MAD111 Chapter 2
15 / 1
The function f : A → B is a bijection (or one-to-one correspondence) if it
is both one-to-one and onto.
TrungDT (FUHN)
MAD111 Chapter 2
15 / 1
The function f : A → B is a bijection (or one-to-one correspondence) if it
is both one-to-one and onto.
TrungDT (FUHN)
MAD111 Chapter 2
15 / 1
The function f : A → B is a bijection (or one-to-one correspondence) if it
is both one-to-one and onto.
TrungDT (FUHN)
MAD111 Chapter 2
15 / 1
TrungDT (FUHN)
MAD111 Chapter 2
16 / 1
Example 1. Which functions are bijection:
TrungDT (FUHN)
MAD111 Chapter 2
16 / 1
Example 1. Which functions are bijection:
(a) f : R → R;
TrungDT (FUHN)
f (x) = x 2
MAD111 Chapter 2
16 / 1
Example 1. Which functions are bijection:
(a) f : R → R;
f (x) = x 2
(b) f : R → R;
f (x) = x 3
TrungDT (FUHN)
MAD111 Chapter 2
16 / 1
Example 1. Which functions are bijection:
(a) f : R → R;
f (x) = x 2
(b) f : R → R;
f (x) = x 3
(c) f : R → Z;
f (x) = b2xc
TrungDT (FUHN)
MAD111 Chapter 2
16 / 1
Example 1. Which functions are bijection:
(a) f : R → R;
f (x) = x 2
(b) f : R → R;
f (x) = x 3
(c) f : R → Z;
f (x) = b2xc
(d) f : Z × Z → Z;
TrungDT (FUHN)
f (m, n) = m + n
MAD111 Chapter 2
16 / 1
Example 1. Which functions are bijection:
(a) f : R → R;
f (x) = x 2
(b) f : R → R;
f (x) = x 3
(c) f : R → Z;
f (x) = b2xc
(d) f : Z × Z → Z;
f (m, n) = m + n
(e) f : Z × Z → Z × Z;
TrungDT (FUHN)
f (m, n) = (m, m + n)
MAD111 Chapter 2
16 / 1
TrungDT (FUHN)
MAD111 Chapter 2
17 / 1
Example 2. Does there exist:
TrungDT (FUHN)
MAD111 Chapter 2
17 / 1
Example 2. Does there exist:
(a) A bijection /one-to-one/onto function from a set of 7 elements to a
set of 5 elements?
TrungDT (FUHN)
MAD111 Chapter 2
17 / 1
Example 2. Does there exist:
(a) A bijection /one-to-one/onto function from a set of 7 elements to a
set of 5 elements? From a set of 5 elements to a set of 7 elements?
TrungDT (FUHN)
MAD111 Chapter 2
17 / 1
Example 2. Does there exist:
(a) A bijection /one-to-one/onto function from a set of 7 elements to a
set of 5 elements? From a set of 5 elements to a set of 7 elements?
(b) A bijection from the set of even integers to the set of odd integers?
TrungDT (FUHN)
MAD111 Chapter 2
17 / 1
Example 2. Does there exist:
(a) A bijection /one-to-one/onto function from a set of 7 elements to a
set of 5 elements? From a set of 5 elements to a set of 7 elements?
(b) A bijection from the set of even integers to the set of odd integers?
(c) A bijection from the set of odd integers to the set of all integers?
TrungDT (FUHN)
MAD111 Chapter 2
17 / 1
Example 2. Does there exist:
(a) A bijection /one-to-one/onto function from a set of 7 elements to a
set of 5 elements? From a set of 5 elements to a set of 7 elements?
(b) A bijection from the set of even integers to the set of odd integers?
(c) A bijection from the set of odd integers to the set of all integers?
(d) A bijection from the set of all real numbers to the set of positive real
numbers?
TrungDT (FUHN)
MAD111 Chapter 2
17 / 1
Compositions and Inverses
TrungDT (FUHN)
MAD111 Chapter 2
18 / 1
Compositions and Inverses
Composition
TrungDT (FUHN)
MAD111 Chapter 2
18 / 1
Compositions and Inverses
Composition
TrungDT (FUHN)
MAD111 Chapter 2
18 / 1
Compositions and Inverses
Composition
TrungDT (FUHN)
MAD111 Chapter 2
18 / 1
TrungDT (FUHN)
MAD111 Chapter 2
19 / 1
Inverse
TrungDT (FUHN)
MAD111 Chapter 2
19 / 1
Inverse
TrungDT (FUHN)
MAD111 Chapter 2
19 / 1
Inverse
Note.
TrungDT (FUHN)
MAD111 Chapter 2
19 / 1
Inverse
Note. The function f : A → B has an inverse if and only if f is a bijection.
TrungDT (FUHN)
MAD111 Chapter 2
19 / 1
2.4 Sequences and Summations
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
Sequence is a discrete structure used to represent an order list.
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
Sequence is a discrete structure used to represent an order list. It is
usually denoted as {a1 , a2 , . . .} = {an , n = 1, 2, . . .}.
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
Sequence is a discrete structure used to represent an order list. It is
usually denoted as {a1 , a2 , . . .} = {an , n = 1, 2, . . .}.
Example.
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
Sequence is a discrete structure used to represent an order list. It is
usually denoted as {a1 , a2 , . . .} = {an , n = 1, 2, . . .}.
Example. Find a general formula for an of each sequence:
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
Sequence is a discrete structure used to represent an order list. It is
usually denoted as {a1 , a2 , . . .} = {an , n = 1, 2, . . .}.
Example. Find a general formula for an of each sequence:
(a)
1 2 3 4
,− , ,− ,...
2 3 4 5
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
Sequence is a discrete structure used to represent an order list. It is
usually denoted as {a1 , a2 , . . .} = {an , n = 1, 2, . . .}.
Example. Find a general formula for an of each sequence:
1 2 3 4
,− , ,− ,...
2 3 4 5
(b) −2, 1, 4, 7, 10, . . .
(a)
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
Sequence is a discrete structure used to represent an order list. It is
usually denoted as {a1 , a2 , . . .} = {an , n = 1, 2, . . .}.
Example. Find a general formula for an of each sequence:
1 2 3 4
,− , ,− ,...
2 3 4 5
(b) −2, 1, 4, 7, 10, . . . (an arithmetic progression)
(a)
(c) 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, . . .
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
Sequence is a discrete structure used to represent an order list. It is
usually denoted as {a1 , a2 , . . .} = {an , n = 1, 2, . . .}.
Example. Find a general formula for an of each sequence:
1 2 3 4
,− , ,− ,...
2 3 4 5
(b) −2, 1, 4, 7, 10, . . . (an arithmetic progression)
(a)
(c) 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, . . .
(d) 1, 1, 2, 3, 5, 8, 13, 21, . . .
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
2.4 Sequences and Summations
Sequences
Sequence is a discrete structure used to represent an order list. It is
usually denoted as {a1 , a2 , . . .} = {an , n = 1, 2, . . .}.
Example. Find a general formula for an of each sequence:
1 2 3 4
,− , ,− ,...
2 3 4 5
(b) −2, 1, 4, 7, 10, . . . (an arithmetic progression)
(a)
(c) 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, . . .
(d) 1, 1, 2, 3, 5, 8, 13, 21, . . . (Fibonacci sequence)
TrungDT (FUHN)
MAD111 Chapter 2
20 / 1
Summations
TrungDT (FUHN)
MAD111 Chapter 2
21 / 1
Summations
Given a sequence {an , n = 1, 2, . . .}.
TrungDT (FUHN)
MAD111 Chapter 2
21 / 1
Summations
Given a sequence {an , n = 1, 2, . . .}. The summation notation,
as follows:
TrungDT (FUHN)
MAD111 Chapter 2
P
, is used
21 / 1
Summations
Given a sequence {an , n = 1, 2, . . .}. The summation notation,
as follows:
100
X
P
, is used
ai = a1 + a2 + · · · + a100
i=1
TrungDT (FUHN)
MAD111 Chapter 2
21 / 1
Summations
Given a sequence {an , n = 1, 2, . . .}. The summation notation,
as follows:
100
X
i=1
100
X
P
, is used
ai = a1 + a2 + · · · + a100
a2i+1 = a3 + a5 + · · · + a201
i=1
TrungDT (FUHN)
MAD111 Chapter 2
21 / 1
Summations
Given a sequence {an , n = 1, 2, . . .}. The summation notation,
as follows:
100
X
i=1
100
X
P
, is used
ai = a1 + a2 + · · · + a100
a2i+1 = a3 + a5 + · · · + a201
i=1
100
X
1 = 1 + 1 + · · · + 1 ( 100 terms )
i=1
TrungDT (FUHN)
MAD111 Chapter 2
21 / 1
Summations
Given a sequence {an , n = 1, 2, . . .}. The summation notation,
as follows:
100
X
i=1
100
X
P
, is used
ai = a1 + a2 + · · · + a100
a2i+1 = a3 + a5 + · · · + a201
i=1
100
X
1 = 1 + 1 + · · · + 1 ( 100 terms )
i=1
Properties
TrungDT (FUHN)
MAD111 Chapter 2
21 / 1
Summations
Given a sequence {an , n = 1, 2, . . .}. The summation notation,
as follows:
100
X
i=1
100
X
P
, is used
ai = a1 + a2 + · · · + a100
a2i+1 = a3 + a5 + · · · + a201
i=1
100
X
1 = 1 + 1 + · · · + 1 ( 100 terms )
i=1
Properties
n
X
i=1
(ai + bi ) =
n
X
i=1
TrungDT (FUHN)
ai +
n
X
bi
i=1
MAD111 Chapter 2
21 / 1
Summations
Given a sequence {an , n = 1, 2, . . .}. The summation notation,
as follows:
100
X
i=1
100
X
P
, is used
ai = a1 + a2 + · · · + a100
a2i+1 = a3 + a5 + · · · + a201
i=1
100
X
1 = 1 + 1 + · · · + 1 ( 100 terms )
i=1
Properties
n
X
i=1
(ai + bi ) =
n
X
i=1
TrungDT (FUHN)
ai +
n
X
i=1
bi
n
X
kai = k
i=1
MAD111 Chapter 2
n
X
ai
i=1
21 / 1
Summations
Given a sequence {an , n = 1, 2, . . .}. The summation notation,
as follows:
100
X
i=1
100
X
P
, is used
ai = a1 + a2 + · · · + a100
a2i+1 = a3 + a5 + · · · + a201
i=1
100
X
1 = 1 + 1 + · · · + 1 ( 100 terms )
i=1
Properties
n
X
i=1
(ai + bi ) =
n
X
i=1
TrungDT (FUHN)
ai +
n
X
i=1
bi
n
X
kai = k
i=1
MAD111 Chapter 2
n
X
ai
i=1
21 / 1
Some Important Sums
TrungDT (FUHN)
MAD111 Chapter 2
22 / 1
Some Important Sums
n
X
i = 1 + 2 + ··· + n =
i=1
TrungDT (FUHN)
n(n + 1)
2
MAD111 Chapter 2
22 / 1
Some Important Sums
n
X
i = 1 + 2 + ··· + n =
i=1
n
X
n(n + 1)
2
i 2 = 12 + 22 + · · · + n2 =
i=1
TrungDT (FUHN)
n(n + 1)(2n + 1)
6
MAD111 Chapter 2
22 / 1
Some Important Sums
n
X
i = 1 + 2 + ··· + n =
i=1
n
X
n(n + 1)
2
n(n + 1)(2n + 1)
6
i=1
n
X
n(n + 1) 2
3
3
3
3
i = 1 + 2 + ··· + n =
2
i 2 = 12 + 22 + · · · + n2 =
i=1
TrungDT (FUHN)
MAD111 Chapter 2
22 / 1
Some Important Sums
n
X
i = 1 + 2 + ··· + n =
i=1
n
X
n(n + 1)
2
n(n + 1)(2n + 1)
6
i=1
n
X
n(n + 1) 2
3
3
3
3
i = 1 + 2 + ··· + n =
2
i=1
n
X
i 2 = 12 + 22 + · · · + n2 =
ri = 1 + r + r2 + · · · + rn =
i=0
TrungDT (FUHN)
1 − r n+1
1−r
MAD111 Chapter 2
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TrungDT (FUHN)
MAD111 Chapter 2
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Example 1.
TrungDT (FUHN)
MAD111 Chapter 2
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Example 1. Find:
TrungDT (FUHN)
MAD111 Chapter 2
23 / 1
Example 1. Find:
(a)
100
X
3i
4i+1
i=1
TrungDT (FUHN)
MAD111 Chapter 2
23 / 1
Example 1. Find:
(a)
(b)
100
X
3i
4i+1
i=1
100
X
i=1
1
(Telescoping sum)
i(i + 1)
Example 2.
TrungDT (FUHN)
MAD111 Chapter 2
23 / 1
Example 1. Find:
(a)
(b)
100
X
3i
4i+1
i=1
100
X
i=1
1
(Telescoping sum)
i(i + 1)
Example 2. Find double summations:
TrungDT (FUHN)
MAD111 Chapter 2
23 / 1
Example 1. Find:
(a)
(b)
100
X
3i
4i+1
i=1
100
X
i=1
1
(Telescoping sum)
i(i + 1)
Example 2. Find double summations:
(a)
2 X
2
X
(i + 2j)
i=1 j=0
TrungDT (FUHN)
MAD111 Chapter 2
23 / 1
Example 1. Find:
(a)
(b)
100
X
3i
4i+1
i=1
100
X
i=1
1
(Telescoping sum)
i(i + 1)
Example 2. Find double summations:
(a)
2 X
2
X
(i + 2j)
i=1 j=0
(b)
10 X
100
X
ij
i=1 j=1
TrungDT (FUHN)
MAD111 Chapter 2
23 / 1