Chapter 4 The Real Number System \(\mathbb{R}\)

4.0.1 Topics Covered

  • Axioms of Real Numbers
  • Order Axioms and Axiom of Bounds
  • Supremum and Infimum of Subsets of \(\mathbb{R}\)
  • Concept of Infinity and Intervals of \(\mathbb{R}\)

4.0.2 What Are the Real Numbers?

The question “What are the real numbers?” is, for now, too deep to answer directly. Instead, we adopt a formal approach: we assume the existence of a set \(\mathbb{R}\), called the set of real numbers, and impose a collection of axioms that define its structure.

All subsequent mathematical development will be based solely on these axioms.

4.1 Axioms of real numbers

We assume that there are two binary operations on \(\mathbb{R}\), that is, “\(+\)” and “\(\cdot\)” the first called addition, and second multiplication, such that \(a + b \in \mathbb{R}\) and \(a\cdot b \in \mathbb{R}\) for all \(a, b \in \mathbb{R}\), and the following properties are satisfied:

A1. Commutativity of Addition
For all \(a, b \in \mathbb{R}\), .\(a + b = b + a\)

A2. Associativity of Addition
For all \(a, b, c \in \mathbb{R}\), \(a + (b + c) = (a + b) + c\)

A3. Existence of Additive Identity
There exists an element \(0 \in \mathbb{R}\) such that for all \(a \in \mathbb{R}\), \(a + 0 = a\)

A4. Existence of Additive Inverse
For every \(a \in \mathbb{R}\), there exists an element \(-a \in \mathbb{R}\) such that: \(a + (-a) = 0\)

A5. Commutativity of Multiplication
For all \(a, b \in \mathbb{R}\), \(a \cdot b = b \cdot a\)

A6. Associativity of Multiplication
For all \(a, b, c \in \mathbb{R}\), \(a \cdot (b \cdot c) = (a \cdot b) \cdot c\)

A7. Existence of Multiplicative Identity
There exists an element \(1 \in \mathbb{R}\) such that for all \(a \in \mathbb{R}\), \(a \cdot 1 = a\)

A8. Existence of Multiplicative Inverse
For every \(a \in \mathbb{R}\) with \(a \ne 0\), there exists an element \(a^{-1} \in \mathbb{R}\) such that: \(a \cdot a^{-1} = 1\)

A9. Distributivity of Multiplication over Addition
For all \(a, b, c \in \mathbb{R}\),\(a \cdot (b + c) = a \cdot b + a \cdot c\)

4.2 Basic Properties of Equality

Let \(x, y, z \in \mathbb{R}\). Then:

  1. Reflexivity:
    \[ x = x \]

  2. Symmetry:
    If \(x = y\), then \(y = x\).

  3. Transitivity:
    If \(x = y\) and \(y = z\), then \(x = z\).

4.3 Some Algebraic Properties of Real Numbers

Let \(a, b, c \in \mathbb{R}\). Then:

  1. Cancellation Law for Addition:
    If \(a + b = a + c\), then \(b = c\).

  2. Multiplication by Zero:
    \[ a \cdot 0 = 0 \]

  3. Zero Product Property:
    If \(a \cdot b = 0\), then \(a = 0\) or \(b = 0\).

Proof.

.

rest of them leave as an exercise

4.3.1 Selected Theorems

These theorems follow from the axioms of the real numbers. In all cases, \(a, b, c, d \in \mathbb{R}\), with additional nonzero conditions where specified.

  1. Properties of Zero

    1. \(a - a = 0\)
    2. \(0 = -a + a\)
    3. \(0 \cdot a = 0\)
    4. If \(ab = 0\), then \(a = 0\) or \(b = 0\)

  2. Properties of Signs

    1. \(-0 = 0\)
    2. \(-(-a) = a\)
    3. \((-a)b = -(ab) = a(-b)\)
    4. \((-a)(-b) = ab\)
    5. \(-a = (-1)a\)

  3. Additional Distributive Properties

    1. \(-(a + b) = -a - b\)
    2. \(a - b = -(b - a)\)
    3. \(-(a - b) = b - a\)
    4. \(a + a = 2a\)
    5. \(a(b - c) = ab - ac = (b - c)a\)
    6. \((a + b)(c + d) = ac + ad + bc + bd\)
    7. \((a + b)(c - d) = ac - ad + bc - bd = (c - d)(a + b)\)
    8. \((a - b)(c - d) = ac - ad - bc + bd\)

  4. Properties of Inverses (for \(a, b \ne 0\))

    1. If \(a \ne 0\), then \(a^{-1} \ne 0\)
    2. \(1^{-1} = 1\)
    3. \((a^{-1})^{-1} = a\)
    4. \((-a)^{-1} = -(a^{-1})\)
    5. \((ab)^{-1} = a^{-1}b^{-1}\)
    6. \(\left( \frac{a}{b} \right)^{-1} = \frac{b}{a}\)

  5. Properties of Quotients (for nonzero denominators)

    1. \(\frac{a}{1} = a\)
    2. \(\frac{1}{a} = a^{-1}\)
    3. \(\frac{a}{a} = 1\)
    4. \(\frac{a/b}{c/d} = \frac{ac}{bd}\)
    5. \(\frac{a/b}{c/d} = \frac{ad}{bc}\)
    6. \(\frac{ac}{bc} = \frac{a}{b}\)
    7. \(\frac{a}{b/c} = \frac{ab}{c}\)
    8. \(\frac{ab}{b} = a\)
    9. \(\frac{-a}{b} = -\left( \frac{a}{b} \right) = \frac{a}{-b}\)
    10. \(\frac{-a}{-b} = \frac{a}{b}\)
    11. \(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\)
    12. \(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\)

Exercise 4.1 Proove theroms

4.4 The Order Axioms and Axiom of Bounds

We assume the existence of a subset \(P \subseteq \mathbb{R}\) with the following properties:

A10. For any \(a \in \mathbb{R}\), exactly one of the following holds: \(a \in P \text{ or } a = 0 \text{ or } -a \in P\)

A11. If \(a, b \in P\), then \(a + b \in P\) and \(a \cdot b \in P\)

Definition 4.1 For any \(a, b \in \mathbb{R}\), we define: \[ a > b \quad \text{if and only if} \quad a - b \in P \]

Equivalently, \(b < a\).

Note: Axioms A10 and A11, together with Axioms A1–A9, endow the field \((\mathbb{R}, +, \cdot)\) with an order structure.
Thus, \(\mathbb{R}\) becomes an ordered field.

Exercise 4.2 Prove that for all \(a \in P\), we have \(a > 0\).

Exercise 4.3 Prove that \(1 \in P\), i.e., \(1 > 0\).

Exercise 4.4 Prove that if \(a > 0\), then \(x_a > 0\) for all \(a \in \mathbb{R}\).

Theorem 4.1 For any two real numbers \(a, b\), exactly one of the following holds: - \(a < b \text{ or } b < a \text{ or } a = b\)

Theorem 4.2 (Order Properties of Real Numbers) Let \(a, b, c \in \mathbb{R}\). Then:

  1. Transitivity
    If \(a < b\) and \(b < c\), then \(a < c\).

  2. Addition Preservation
    If \(a < b\), then \(a + c < b + c\).

  3. Multiplication by Positive
    If \(c > 0\) and \(a < b\), then \(a \cdot c < b \cdot c\).

  4. Multiplication by Negative
    If \(c < 0\) and \(a < b\), then \(a \cdot c > b \cdot c\).
    (Note: This reverses the inequality.)

4.5 Absolute Value

Definition 4.2 For any real number \(x\), the absolute value (or modulus) of \(x\) is defined as: \[ |x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases} \]

Theorem 4.3 (Properties of Absolute Value) Let \(a, b \in \mathbb{R}\). Then:

  1. \(|a| = 0 \iff a = 0\)
  2. \(|a| \geq 0\)
  3. \(|a| = |-a|\)
  4. \(-|a| \leq a \leq |a|\)
  5. If \(b > 0\), then \(|a| \leq b \iff -b \leq a \leq b\)
  6. If \(b > 0\), then \(|a| \geq b \iff a \geq b\) or \(a \leq -b\)
  7. \(|a + b| \leq |a| + |b|\)

Proof: Leave as a exercise

Example 4.1

  1. If \(a < b\) and both \(a, b > 0\), then \(a^{-1} > b^{-1}\).
  2. \(ab > 0 \iff a, b\) are both positive or both negative.
  3. \(ab < 0 \iff\) one of \(a, b\) is positive and the other is negative.

Definition 4.3 (Minimum and Maximum) Let \(A \subseteq \mathbb{R}\) be a non-empty set.

  • An element \(m \in A\) is called the minimum of \(A\) if:\(m \leq x \quad \text{for all } x \in A\)

  • An element \(M \in A\) is called the maximum of \(A\) if:\(M \geq x \quad \text{for all } x \in A\)

Note: If a minimum or maximum exists, it is unique.

Example 4.2 Let \(A = \{2, -3, 8, 17, -5\}\)

  • Minimum of \(A\):
    \(\min A = -5\). (Because \(2 \geq -5, -3\geq -5, 8 \geq -5, 17\geq -5, -5 \geq -5\)).

  • Maximum of \(A\):
    \(\max A = 17\). (Because \(2 \leq 17, -3\leq 17, 8 \leq 17, 17\leq 17, -5 \leq 17\))

Example 4.3 Let \(A = \{1 + \frac{2}{n} \mid n \in \mathbb{N} \}\)

This is an infinite set of real numbers: \[ A = \left\{1 + \frac{2}{1}, 1 + \frac{2}{2}, 1 + \frac{2}{3}, \dots \right\} = \left\{3, 2, \frac{5}{3}, \dots \right\} \]

Definition 4.4 (Bounded Above) A non-empty set \(A \subseteq \mathbb{R}\) is said to be bounded above if there exists a number \(k \in \mathbb{R}\) such that:\(\forall x \in A, \quad x \leq k\)

The number \(k\) is called an upper bound of the set \(A\).

Definition 4.5 (Bounded Below) A non-empty set \(A \subseteq \mathbb{R}\) is said to be bounded below if there exists a number \(k \in \mathbb{R}\) such that:

\[ \forall x \in A, \quad x \geq k \]

The number \(k\) is called a lower bound of the set \(A\).

Note: Let \(M \subseteq d\) and \(A \subseteq \mathbb{R}\).

  • A set \(A\) is bounded above \(\underbrace{\iff}_{\text{Def}^\text{n}} \exists k \in \mathbb{R} \quad \forall x \in A, \quad x \leq k\)

  • A set \(A\) is not bounded above \(\iff \neg (\exists k \in \mathbb{R} \quad \forall x \in A, \quad x \leq k)\equiv\forall k \in \mathbb{R} \quad \exists x \in A, \quad x > k\)

Example 4.4 Let \(A = \{ x \in \mathbb{R} \mid x \geq -7 \}\) and \(B = \{ 2n + 3 \mid n \in \mathbb{N} \}\)

Note: Let \(M \subseteq d\) and \(A \subseteq \mathbb{R}\).

  • A set \(A\) is bounded below \(\underbrace{\iff}_{\text{Def}^\text{n}} \exists k \in \mathbb{R} \quad \forall x \in A, \quad x \geq k\)

  • A set \(A\) is not bounded below \(\iff \neg (\exists k \in \mathbb{R} \quad \forall x \in A, \quad x \geq k)\equiv\forall k \in \mathbb{R} \quad \exists x \in A, \quad x > k\)

Definition 4.6 Let \(A \subseteq \mathbb{R}\), with \(A \neq \emptyset\).

  • A set \(A\) is bounded if it is both bounded above and bounded below.
  • A set \(A\) is unbounded if it is not bounded.

Definition 4.7 (Least Upper Bound (Supremum)) An element \(\lambda \in \mathbb{R}\) is called the least upper bound (or supremum) of \(A\) if:

  1. \(\lambda\) is an upper bound of \(A\), i.e., \[ \forall x \in A, \quad x \leq \lambda \]
  2. No upper bound of \(A\) is less than \(\lambda\), i.e., \[ \forall u \in \mathbb{R}, \left( \forall x \in A, \ x \leq u \right) \Rightarrow \lambda \leq u \]

Definition 4.8 (Greatest Lower Bound (Infimum)) An element \(\mu \in \mathbb{R}\) is called the greatest lower bound (or infimum) of \(A\) if:

  1. \(\mu\) is a lower bound of \(A\), i.e., \[ \forall x \in A, \quad x \geq \mu \]
  2. No lower bound of \(A\) is greater than \(\mu\), i.e., \[ \forall u \in \mathbb{R}, \left( \forall x \in A, \ x \geq u \right) \Rightarrow \mu \geq u \]

4.6 Axiom of Bound (Completeness Axiom)

A12. Least Upper Bound Axiom

If \(A \subseteq \mathbb{R}\) is non-empty and bounded above, then:

\[ \exists \sup A \in \mathbb{R} \]

That is, every non-empty subset of \(\mathbb{R}\) that is bounded above has a least upper bound (supremum) in \(\mathbb{R}\).

A12′. Greatest Lower Bound Axiom

If \(A \subseteq \mathbb{R}\) is non-empty and bounded below, then:

\[ \exists \inf A \in \mathbb{R} \]

That is, every non-empty subset of \(\mathbb{R}\) that is bounded below has a greatest lower bound (infimum) in \(\mathbb{R}\).

Example 4.5 Let \(A = \{ x \in \mathbb{R} \mid x^2 < 2 \}\)

  • This set is bounded above.
  • The least upper bound (supremum) of \(A\) is:

\[ \sup A = \lambda \]

Since:

\[ \sqrt{2}^2 = 2 \quad \text{and} \quad \sqrt{2} \in \mathbb{R} \]

4.7 Archimedean Property

Theorem 4.4 (Archimedean Property) For all \(a \in \mathbb{R}^+\) and \(b \in \mathbb{R}\), there exists \(n \in \mathbb{N}\) such that: \(na \geq b\)

Proof. Suppose the Archimedean Property is false. Then\(\exists a \in \mathbb{R}^+, \exists b \in \mathbb{R}, \forall n \in \mathbb{N}, \quad na \leq b\)

Define the set \(S := \{ na \mid n \in \mathbb{N} \}\)

Then \(b\) is an upper bound of \(S\). By the completeness axiom, the supremum \(s_0 := \sup S\) exists.

Let \(n \in \mathbb{N}\). Then \(n + 1 \in \mathbb{N}\), and \(s_0 \geq (n + 1)a = na + a \Rightarrow s_0 - a \geq na\)

So \(s_0 - a\) is an upper bound of \(S\), but \(s_0 - a < s_0\)

This contradicts the fact that \(s_0\) is the least upper bound of \(S\). Hence, the Archimedean Property holds. 

Corollary 4.1 For any \(x \in \mathbb{R}\) with \(x > 0\), there exists \(n \in \mathbb{N}\) such that \(\frac{1}{n} < x\)

Proof. Let \(a = x\), \(b = 1\). Since \(a > 0\), by the Archimedean Property, there exists \(n \in \mathbb{N}\) such that:

\[ na > 1 \Rightarrow x > \frac{1}{n} \]

4.8 Concept of Infinity and Intervals of \(\mathbb{R}\)

4.8.1 Concept of Infinity

The infinity symbol is a mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate,
The infinity symbol is a mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate,

Infinity comes from the Latin infinitas, meaning “unboundedness.” It refers to several distinct concepts—usually linked to the idea of “without end.”

  • We encounter infinity for the first time in school, where we meet the set:

\[ \mathbb{N} = \{1, 2, 3, \dots\} \]

  • The concept of this set can be formulated as follows:

For each natural number \(n\), there exists a larger natural number \(n + 1\).
In other words, there does not exist a natural number that is larger than all natural numbers.
That is, there exists no largest natural number.

• We are unable to write down the list of all natural numbers. • Hence, our writing is never- ending process, because of this we talk about infinity or unboundedness of natural numbers. • We realize that we have infinitely many natural numbers, but we are unable to perceive all natural numbers at once. • We have a similar situation with the idea (the notion) of a line in Geometry. • Length of any line is infinite or unbounded (infinitely large) • One can walk along a line for an arbitrarily long time and one never reaches the end. • We unable to see a whole infinite line at once.

• The main problem with understanding the concept of infinity is that we are not capable of imagining object at once. • We are only able to see a finite fraction (part) of an infinite object. • The way out we use is to denote infinite objects by symbols. • We work with these symbols as a finite representation of the corresponding infinite objects. Infinity is not a number, but a concept that describes something unbounded or without limit.

  • Infinity is not a number, but a concept that describes something unbounded or without limit.

  • In mathematics, we denote infinity using the symbol:

\[ \infty \]

4.8.2 Infinity in \(\mathbb{R}\)

Let \(S \subseteq \mathbb{R}\) be a non-empty subset.

  • \(S\) is not bounded above if it has no upper bound.
  • \(S\) is not bounded below if it has no lower bound.

To facilitate mathematical reasoning, we adjoin two fictitious points to \(\mathbb{R}\):

  • \(+\infty\) (often written simply as \(\infty\))
  • \(-\infty\)

These points are not elements of \(\mathbb{R}\), i.e.,

\[ \infty, -\infty \notin \mathbb{R} \]

For any real number \(x \in \mathbb{R}\):

\[ -\infty < x < \infty \]

If \(S \subseteq \mathbb{R}\) is non-empty and not bounded above, we write:

\[ \sup S = \infty \]

This notation indicates that \(S\) has no finite supremum.

4.8.3 Intervals in \(\mathbb{R}\)

Definition 4.9 Let \(a, b \in \mathbb{R}\) with \(a < b\). We define the following intervals:

  • Bounded Intervals

    • Closed Interval: \[ [a, b] = \{ x \in \mathbb{R} \mid a \leq x \leq b \} \]

    • Open Interval: \[ (a, b) = \{ x \in \mathbb{R} \mid a < x < b \} \]

    • Half-Open Intervals: \[ (a, b] = \{ x \in \mathbb{R} \mid a < x \leq b \}, \quad [a, b) = \{ x \in \mathbb{R} \mid a \leq x < b \} \]

These are called bounded intervals because both endpoints \(a\) and \(b\) are real numbers.

  • Unbounded Intervals

    • Right-Unbounded: \[ [a, \infty) = \{ x \in \mathbb{R} \mid x \geq a \}, \quad (a, \infty) = \{ x \in \mathbb{R} \mid x > a \} \]

    • Left-Unbounded: \[ (-\infty, b] = \{ x \in \mathbb{R} \mid x \leq b \}, \quad (-\infty, b) = \{ x \in \mathbb{R} \mid x < b \} \]

    • Entire Real Line: \[ (-\infty, \infty) = \mathbb{R} \]

  • Non-Emptiness of Open Intervals

    Any open interval \((a, b)\) with \(a < b\) is nonempty. For example, it contains \(\frac{a + b}{2}\)

Example 4.6 Write Down a Set

  1. Set \(A\) such that \(\sup A = \infty\), \(\inf A = 2\): \[ A = [2, \infty) \]

  2. Set \(B\) such that \(\sup B = \infty\), \(\inf B = 1\): \[ B = [1, \infty) \]

  3. Set \(C\) such that \(\sup C = 2\), \(\inf C = -\infty\): \[ C = (-\infty, 2] \]

  4. Set \(D\) such that \(\sup D = \infty\), \(\inf D = -\infty\): \[ D = (-\infty, \infty) = \mathbb{R} \]