Real Analysis (1)

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Real Analysis (1)

Preliminaries

Sets

If an element \(x\) is in a set \(A\), we write:

\[x \in A\]

and say that \(x\) is a member of \(A\), or that \(x\) belongs to \(A\). If \(x\) is not in \(A\), we write:

\[x \notin A\]

If every element of a set \(A\) also belongs to a set \(B\), we say that \(A\) is a subset of \(B\) and write:

\[A \subseteq B\]

We say that \(A\) is a proper subset of \(B\) if \(A \subset B\), but there is at least one element of \(B\) that is not in \(A\). In this case we write:

\[A \subset B\]

Definition 1.11: Equal Sets

Two sets \(A\) and \(B\) are said to be equal, we write \(A = B\) if they contain the same elements. Thus, to prove that sets \(A\) and \(B\) are equal, we must show that:

\[A \subseteq B, \;\; B \subseteq A\]

A set is normally defined by either listing its elements explicitly, or by specifying a property that determines the elements of the set. If \(P\) denotes a property that is meaningful and unambiguous for elements of a set \(S\), then we write:

\[\{x \in S; P(x)\}\]

for the set of all elements \(x\) in \(S\) for which the property \(P\) is true.

De Morgan's Law

If \(A, B, C\) are sets, then:

\[A / (B \cup C) = (A / B) \cap (A / C)\] \[A / (B \cap C) = (A / B) \cup (A / C)\]

Where \(A / B\) denotes the complement of \(B\) relative to \(A\):

\[A / B := \{x : x \in A \cap x \notin B\}\]

Unions and Interactions

For a finite collection of sets \(\{A_1,...., A_n\}\), their union is the set \(A\) consisting of all elements that belong to at least one of the sets \(A_k\), and their intersection consists of all elements that belong to all of the sets \(\{A_k\}\).

For an infinite collection of sets \(\{A_1, ...., A_n, ...\}\), their union is the set of elements that belong to at least one of the sets \(A_k\):

\[\bigcup^{\infty}_{n=1} A_n = \{x : \exists \; n \in \mathbb{N}, \text{s.t } x \in A_n \}\]

Similarly, their intersection is the set of elements that belong to all of these sets \(A_n\):

\[\bigcap^{\infty}_{n=1} A_n = \{x : \forall \; n \in \mathbb{N}, \text{s.t } x \in A_n \}\]

Functions

Definition 1.15: Cartesian Product

If \(A\) and \(B\) are nonempty sets, then the Cartesian product \(A \times B\) of \(A\) and \(B\) is the set of all ordered pairs \((a, b)\) with \(a \in A\) and \(b \in B\). That is,

\[A \times B := \{(a, b): a \in A, b \in B\}\]

General Definition

A function \(f\) from a set \(A\) into a set \(B\) is a rule of correspondence that assigns to each element \(x \in A\) an uniquely determined element \(f(x)\) in \(B\).

Definition 1.1.6: Function as sets

Let \(A\) and \(B\) be sets. Then a function from \(A\) to \(B\) is a set \(f\) of ordered pairs in \(A \times B\) such that for each \(a \in A\) there exists a unique \(b \in B\) with \((a, b) \in f\)

The set \(A\) of first elements of a function \(f\) is called the domain of \(f\) and is often denoted by \(D(f)\). The set of all second elements in \(f\) is called the range of \(f\) and denoted as \(R(f)\). Notice that:

\[D(f) = A\]

but:

\[R(f) \subseteq B\]

The notation:

\[f: A \rightarrow B\]

is often used to indicate that \(f\) is a function from \(A\) into \(B\). We will also say that \(f\) is a mapping of \(A\) into \(B\). If \(b = f(a)\), we often refer to \(b\) as the value of \(f\) at \(a\), or as the image of \(a\) under \(f\).

Direct and Inverse Images

Let \(f: A \rightarrow B\) be a function with domain \(D(f) = A\) and range \(R(f) \subseteq B\)

Definition 1.1.7: Direct Image and Inverse Image

If \(E\) is a subset of \(A\), then the direct image of \(E\) under \(f\) is the subset \(f(E)\) of \(B\) given by:

\[f(E):= \{f(x): x \in E\}\]

If \(H\) is a subset of \(B\), then the inverse image of \(H\) under \(f\) is the subset \(f^{-1}(H)\) of \(A\) given by:

\[f^{-1}(H) := \{x \in A: f(x) \in H\}\]

Definition 1.1.9: Injection, Surjection and Bijection

Let \(f: A \rightarrow B\) be a function from \(A\) to \(B\)

  1. The function \(f\) is said to be Injective(one to one) if whenever \(x_1 \neq x_2\), then \(f(x_1) \neq f(x_2)\).
  2. The function \(f\) is said to be Surjective(to map \(A\) onto \(B\)) if whenever \(f(A) = B\), that is if \(R(f) = B\).
  3. The function \(f\) is said to be Bijective if \(f\) is both injective and surjective.

Definition 1.1.11: Inverse Functions

If \(f: A \rightarrow B\) is bijection of \(A\) onto \(B\) then:

\[g:= \{(b, a) \in B \times A: (a, b) \in f\}\]

is a function on \(B\) into \(A\). This function is called inverse function of \(f\) and denoted as \(f^{-1}\)

Remark, \(f^{-1}\) (inverse image) make sense even if \(f\) has no inverse function. However, if the inverse function \(f^{-1}\) does exist, then \(f^{-1} (H)\) is the direct image of the set \(H \subseteq B\) under \(f^{-1}\)

Definition 1.1.12: Composition of Functions

If \(f: A \rightarrow B\) and \(g : B \rightarrow C\), and if \(R(f) \subseteq D(g) = B\), then the composition function \(g \circ f\) is the function from \(A \rightarrow C\):

\[(g \circ f) (x) := g(f(x))\]

Theorem 1.1.14

Let \(f: A \rightarrow B\) and \(g: B \rightarrow C\) be functions and let \(H\) be a subset of \(C\), then we have:

\[(g \circ f)^{-1} (H) = f^{-1}(g^{-1} (H))\]

Restrictions of Functions

If \(f: A \rightarrow B\) is a function and if \(A_1 \subset A\), we can define a function \(f_1: A_1 \rightarrow B\) by:

\[f_1(x) := f(x) \;\; x \in A_1\]

The function \(f_1\) is called the restriction of \(f\) to \(A_1\) denoted by \(f_1 = f|A_1\).

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

  1. The empty set \(\emptyset\) is said to have 0 elements.
  2. If \(n \in \mathbb{N}\), a set \(S\) is said to have \(n\) elements if there exists a bijection from the set \(\mathbb{N}: \{1, 2, ...., n\}\) onto S.
  3. A set \(S\) is said to be finite, if it is either empty or it has \(n\) elements for some \(n \in \mathbb{N}\).
  4. A set \(S\) is said to be infinite if it is not finite.
  5. A set \(T_1\) is finite if and only if there is a bijection from \(T_1\) onto another set \(T_2\) that is finite.

Uniqueness Theorem:

If \(S\) is a finite set, then the number of elements in \(S\) is a unique number in \(\mathbb{N}\).

Theorem 1.3.3

The set \(\mathbb{N}\) of natural numbers is an infinite set.

Theorem 1.3.4

  1. If \(A\) is a set with \(m\) elements and \(B\) is a set with \(n\) elements and if \(A \cap B = \emptyset\), then \(A \cup B\) has \(m + n\) elements.
  2. If \(A\) is a set with \(m \in \mathbb{N}\) elements and \(C \subseteq A\) is a set with 1 element, then \(A/ C\) is a set with \(m-1\) elements.
  3. If \(C\) is an infinite set and \(B\) is a finite set, then \(C/ B\) is an infinite set.

Theorem 1.3.5

Suppose that \(S\) and \(T\) are sets and that \(T \subseteq S\):

  1. If \(S\) is a finite set, then \(T\) is a finite set.
  2. If \(T\) is an infinite set, then \(S\) is an infinite set.

Definition 1.3.6: Denumberable, Countable

  1. A set \(S\) is said to be denumberable (countably infinite) if there exists a bijection of \(\mathbb{N}\) onto \(S\).
  2. A set \(S\) is said to be countable if it is either finite or denumberable.
  3. A set \(S\) is said to be uncountable if it is not countable.

Theorem 1.3.8

The set \(\mathbb{N} \times \mathbb{N}\) is denumberable

Theorem 1.3.9

Suppose that \(S\) and \(T\) are sets and that \(T \subset S\):

  1. If \(S\) is a countable set, then \(T\) is a countable set.
  2. If \(T\) is an uncountableset, then \(S\) is an uncountable sets.

Theorem 1.3.10

The following statements are equivalent:

  1. \(S\) is a countable set.
  2. There exists a surjection of \(\mathbb{N}\) onto \(S\).
  3. There exists an injection of \(S\) into \(\mathbb{N}\).

Theorem

The set \(\mathbb{Q}\) of all rational numbers is denumberable.

Theorem 1.3.12

If \(A_m\) is a countable set for each \(m \in \mathbb{N}\), then the union \(A:= \bigcup^{\infty}_{m=1} A_m\) is countable.

Real Numbers

Least Upper Bounds (Supremum) and Greatest Lower Bounds (infimum)

Definition 1.1.3.1

A set \(A \subseteq \mathbb{R}\) is bounded above if there exists a number \(b \in \mathbb{R}\) such that \(a \leq b\) for all \(a \in A\). The number \(b\) is called an upper bound for \(A\). Similarly, the set \(A\) is bounded below if there exists a lower bound \(l \in \mathbb{R}\) satisfying \(l \leq a\) for every \(a \in A\).

Definition 1.1.3.2

A real number \(s\) is the least upper bound for a set \(A \subseteq \mathbb{R}\) if:

  1. \(s\) is an upper bound for \(A\).
  2. If \(b\) is any upper bound for \(A\), \(b \geq s\).

The least upper bound is also called supremum of the set \(A\) denoted as:

\[\sup A\]

The greatest lower bound is called infimum for \(A\):

\[\inf A\]

infimum and supremum are unique for a set \(A\) and may or may not be elements in the set \(A\).

Definition 1.1.3.4

A real number \(a_0\) is a maximum of the set \(A\) if \(a_0\) is an element of \(A\) and \(a_0 \geq a\) for all \(a \in A\). Similarly, a number \(a_1\) is a minimum of \(A\) if \(a_1 \in A\) and \(a_1 \leq a\) for every \(a \in A\).

Axiom of Completeness

Every nonempty set of real numbers that is bounded above has a least upper bound.

Lemma 1.1.3.8

Assume \(s \in \mathbb{R}\) is an upper bound for a set \(A \subseteq \mathbb{R}\). Then \(s = \sup A\) if and only if for every choice of \(\epsilon > 0\), there exists an element \(a \in A\) satisfying \(s < a + \epsilon\).

Theorem 1.1.4.1: Nested Interval Property

For each \(n \in \mathbb{N}\), assume we are given a closed interval \(I_n = [a_n, b_n] = \{x \in \mathbb{R}: a_n \leq x \leq b\}\). Assume also that each \(I_n\) contains \(I_{n+1}\). Then the resulting nested sequence of closed intervals:

\[I_1 \supseteq I_2 \supseteq I_3 .... \]

has a nonempty intersection. That is:

\[\bigcap^{\infty}_{n=1} I_{n} \neq 0\]

Theorem 1.1.4.2: Archimedean Property

  1. Given any number \(x \in \mathbb{R}\), there exists an \(n \in \mathbb{N}\) satisfying \(n > x\).
  2. Given any real number \(y > 0\), there exists an \(n \in \mathbb{N}\) satisfying \(\frac{1}{n} < y\)

Theorem 1.1.4.3: Density of \(\mathbb{Q}\) in \(\mathbb{R}\)

For every two real numbers \(a\) and \(b\) with \(a < b\), there exists a rational number \(r\) satisfying \(a < r < b\). In other words, rational numbers are dense in \(\mathbb{R}\).

Theorem 1.1.4.5

There exists a real number \(\alpha \in \mathbb{R}\) satisfying \(\alpha^2 = x, \;\; \forall x \in \mathbb{R} \geq 0\)

Cantor's Diagonalizing Method

Theorem 1.1.6.1

The open interval \((0, 1) = \{x \in \mathbb{R}: 0 < x < 1\}\) is uncountable

Power Sets

Given a set \(A\), the power set \(P(A)\) refers to the collection of all subsets of \(A\). It is important to understand that \(P(A)\) is itself considered a set whose elements are the different possible subsets of \(A\).

Cantor's Theorem

Given any set \(A\), there does not exist a function \(f: A \rightarrow P(A)\) that is onto. That is, no function satisfies \(R(f) = B\).

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