Real Analysis (1)
Real Analysis (1)
Preliminaries
Sets
If an element
and say that
If every element of a set
We say that
Definition 1.11: Equal Sets
Two sets
A set is normally defined by either listing its elements explicitly, or by specifying a property that determines the elements of the set. If
for the set of all elements
De Morgan's Law
If
Where
Unions and Interactions
For a finite collection of sets
For an infinite collection of sets
Similarly, their intersection is the set of elements that belong to all of these sets
Functions
Definition 1.15: Cartesian Product
If
General Definition
A function
Definition 1.1.6: Function as sets
Let
The set
but:
The notation:
is often used to indicate that
Direct and Inverse Images
Let
Definition 1.1.7: Direct Image and Inverse Image
If
If
Definition 1.1.9: Injection, Surjection and Bijection
Let
- The function
is said to be Injective(one to one) if whenever , then . - The function
is said to be Surjective(to map onto ) if whenever , that is if . - The function
is said to be Bijective if is both injective and surjective.
Definition 1.1.11: Inverse Functions
If
is a function on
Remark,
Definition 1.1.12: Composition of Functions
If
Theorem 1.1.14
Let
Restrictions of Functions
If
The function
Finite and Infinite Sets
From a mathematical perspective, what we are doing is defining a bijective mapping between the set and a portion of the set of natural numbers. If the set is such that the counting does not terminate, such as the set of natural numbers itself, then we describe the set as being infinite.
Definition 1.3.1: Set Elements, finite, infinite
- The empty set
is said to have 0 elements. - If
, a set is said to have elements if there exists a bijection from the set onto S. - A set
is said to be finite, if it is either empty or it has elements for some . - A set
is said to be infinite if it is not finite. - A set
is finite if and only if there is a bijection from onto another set that is finite.
Uniqueness Theorem:
If
Theorem 1.3.3
The set
Theorem 1.3.4
- If
is a set with elements and is a set with elements and if , then has elements. - If
is a set with elements and is a set with 1 element, then is a set with elements. - If
is an infinite set and is a finite set, then is an infinite set.
Theorem 1.3.5
Suppose that
- If
is a finite set, then is a finite set. - If
is an infinite set, then is an infinite set.
Definition 1.3.6: Denumberable, Countable
- A set
is said to be denumberable (countably infinite) if there exists a bijection of onto . - A set
is said to be countable if it is either finite or denumberable. - A set
is said to be uncountable if it is not countable.
Theorem 1.3.8
The set
Theorem 1.3.9
Suppose that
- If
is a countable set, then is a countable set. - If
is an uncountableset, then is an uncountable sets.
Theorem 1.3.10
The following statements are equivalent:
is a countable set.- There exists a surjection of
onto . - There exists an injection of
into .
Theorem
The set
Theorem 1.3.12
If
Real Numbers
Least Upper Bounds (Supremum) and Greatest Lower Bounds (infimum)
Definition 1.1.3.1
A set
Definition 1.1.3.2
A real number
is an upper bound for .- If
is any upper bound for , .
The least upper bound is also called supremum of the set
The greatest lower bound is called infimum for
infimum and supremum are unique for a set
Definition 1.1.3.4
A real number
Axiom of Completeness
Every nonempty set of real numbers that is bounded above has a least upper bound.
Lemma 1.1.3.8
Assume
Theorem 1.1.4.1: Nested Interval Property
For each
has a nonempty intersection. That is:
Theorem 1.1.4.2: Archimedean Property
- Given any number
, there exists an satisfying . - Given any real number
, there exists an satisfying
Theorem 1.1.4.3: Density of in
For every two real numbers
Theorem 1.1.4.5
There exists a real number
Cantor's Diagonalizing Method
Theorem 1.1.6.1
The open interval
Power Sets
Given a set
Cantor's Theorem
Given any set