(2) MODULE 1: Properties of Real Numbers

Reflexivity for Equality

a = a

Symmetry for Equality

if a = b then b = a

Transitivity for Equality

if a = b and b = c then a = c

Addition Property for Equality

if a = b then a + c = b + c

Multiplication Property for Equality

if a = b then a · c = b · c

Real Number System

It consists of the set of real number and two operations, addition (+) and multiplication (·).

1. Reflexivity for Equality

2. Symmetry for Equality

3. Transitivity for Equality

4. Addition Property for Equality

5. Multiplication Property for Equality

What are the 5 Equality Axioms ?

1. Closure

2. Associativity

3. Commutativity

4. Distributivity of Multiplication over Addition

5. Existence of Identity Elements

6. Existence of Inverses

What are the 6 Field Axioms ?

Closure

• Addition: The sum of two real numbers a and b, denoted by a + b, is also a real number.

• Multiplication: The product of two real numbers a and b, denoted by a · b, is also a real number.

Associativity

• Addition: (a + b) + c = a + (b + c)

• Multiplication: (a · b) · c = a · (b · c)

Commutativity

• Addition: a + b = b + a

• Multiplication: a · b = b · a

Distributivity of Multiplication over Addition

• Left Hand: c · (a + b) = (c · a) + (c · b)

• Right Hand: (a + b) · c = (a · c) + (b · c)

Existence of Identity Elements

• Additive Identity: There exists a unique number 0 such that a + 0 = a.

• Multiplicative Identity: There exists a unique number 1 such that a · 1 = a.

Existence of Inverses

• Additive Inverse: There exists a unique number −a (read as “negative of a”) for any real number a such that a + (−a) = 0.

• Multiplicative Inverse: There exists a unique number 1/a for any real number a does not equal to 0 such that a · 1/a = 1.

1. Trichotomy Axiom

2. Transitivity for Inequality

3. Addition Rule for Inequalities

4. Multiplication Rule for Inequalities

What are the 4 Order Axioms ?

Closure of Positive Numbers

• Addition: If a > 0 and b > 0 then a + b > 0.

• Multiplication: If a > 0 and b > 0 then ab > 0.

One-Dimensional Coordinate System

It is a geometric interpretation of real numbers. Each real number is associated with a point along a horizontal line to form the real number line.

1. Open Interval

2. Closed Interval

3. Half Open (Half Closed) Interval

4. Open Ray Interval

5. Closed Ray Interval

The 5 Types of Interval Notation

Inequality

It is a statement saying that one expression is less than or equal to another.

Domain

The set of real numbers for which the members if the inequality is defined is the ?

Solutions of Inequality

These admissible values of the variable, if any that result in a true statement are called ?

Conditional inequality

It is an inequality that is true only for some values, that is there exists at least one element of the domain which is not part of the solution set.

Addition Rule for Inequalities

If a > b then a + c > b + c.

Multiplication Rule for Inequalities

If a > b and c > 0 then ac > bc.

Critical numbers

These are numbers that can either make the entire expression be equal to zero or make the expressions undefined.