# MATH 409 Lecture 1

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**Begin Exam 1 content**

Office appointments: Generally available for TWR afternoons

Topics:

- Axioms
- Point Set Theory
- Compactness, completeness, and connectedness
- Continuity and Uniform Continuity
- Sequencies, series
- Differentiatiability
- Theory of Riemann integration

## Challenge 1

50 pts. No deadline

Let be an infinitely differentiable function. Suppose that for any point there iexists a derivative of that vanishes at :

Prove that is polynomial.

**Note:**A polynomial can be uniquely characterized as an infinitely differentiable function such that (identically zero) for some .

### Smaller Challenge

What is the name of the river on the cover of the book.

## Axioms of an Ordered Field

### Real Line

Study of calculus of functions begins with study of domain: real numbers (real line)

The real line is a mathematical object rich with structure:

- algebraic structure (4 operations: add, subtract, multiplication, and division)
- ordering (when choosing any 3 points, one is located between the other two
- metric structure (measurable distance between points)
- continuity (we can get from one point to another in a continuous way)

#### Axiomatic Model

Provides solid foundation for all subsequent developments

Three **postulates**, each consisting of one or several axioms

To verify adequacy, prove that axioms are **consistent** (i.e. there exists an object satisfying them), and **categorical** (i.e. object is, in a sense, unique)

Axioms chosen among basic properties of real numbers:

- Formalizes algebraic structure
- Formalizes ordering
- Formalizes continuous structure

**Note:**Metric structure can be formaziled in terms of other structures

### Field

Motivated by the real numbers and complex numbers

Informally, a field is a set with 4 arithmetic operations (+, −, ×, ÷) that have roughly the same properties as those of real (or complex) numbers.

Notion of field is important for linear algebra. Members of a field can serve as a set of scalars for a vector space.

Formally, A **field** is a set equipped with two closed binary operations:

*addition:**multiplication:*

Which adhere to the following axioms:

F1. | for all | commutativity of addition |
---|---|---|

F2. | for all | associativity of addition |

F3. | There exists an element of , denoted , such that for all | additive identity |

F4. | For any , there exists an element of , denoted , such that | additive inverse |

F1'. | for all | commutativity of multiplication |

F2'. | for all | associativity of multiplication |

F3'. | There exists an element of different from , denoted , such that for all | multiplicative identity |

F4'. | For any , , there exists an element of , denoted , such that | multiplicative inverse |

F5. | for all | distributive property |

Subtraction and division are then defined as compound operations:

**Postulate 1:**The set of real numbers is a field.

Other examples of fields include:

- complex numbers
- rational numbers
- Rational functions in variable with real coefficients (e.g. for and )
- Field of two elements

#### Properties

**Theorem.** The zero 0 is unique.

*Proof.* Suppose and are both zeroes, so and for all .

Then and . Hence .

**Theorem.** For any , the negative is unique.

*Proof.* Suppose and are both negatives of . Let's compute the sum of in two ways:

By associativity of addition, .

**Theorem. [Cancellation Law].** implies for any

*Proof.* If then . By associativity, and .

Hence .

**Theorem.** for any

*Proof.* Start with . Apply the distributive law to get

Let's write this as . By the cancellation law,

**Theorem.** for any

*Proof.* There are two ways to add: . This is equal to by the previous property.

Other properties:

- for all
- for all
- The unity 1 is unique.
**(same proof as uniqueness of 0)** - For any , the inverse is unique.
- (Cancellation law) implies whenever .
- For any , the equality implies that either or .

## Relations

Recall that the Cartesian product of two sets and is the set of all ordered pairs such that and .

The Cartesian square is denoted

A **relation** on a set is (identified with) a subset of its Cartesian square .

If then we say ** is related to ** (in the sense of or by ) and write .

Examples:

- Equality:
- Not equal to: (complement of equality)
- Less than:
- Less than or equal to:
- Is contained in:
- Divides: for

### Properties

Let be a relation on a set

**Reflexive:**for all**Symmetric:**implies**Antisymmetric:**and cannot hold simultaneously**Weakly antisymmetric:**and imply that**Transitive:**for all , , imply

### Partial Ordering

Relation on a set is a **partial ordering** (or **partial order**, or simply **order**) if is reflexive, weakly antisymmetric, and transitive:

- and imply
- and imply

(e.g. less than or equal to)

#### Strict Partial Ordering

A relation on a set is a **strict partial order** (or **strict order**) if is antisymmetric and transitive

- implies
- and imply

(e.g. less than)

### Linear / Total Ordering

An order on a set is called **linear** (or **total**) if for any elements at least one of the following statements hold:

- , or

**Postulate 2:**There is a relation on the set of real numbers , denoted , that is a strict linear order. Moreover, this order and arithmetic operations on satisfy the following axioms:

OA. | implies for all |
---|---|

OM1. | and imply for all |

OM2. | and imply for all |

the axioms OM1 and OM2 can be replaced by a third | |

OM. | and imply for all |

#### Auxiliary notation

- means that
- by , we mean that or .
- By , we mean that and

#### Properties of Linearly Ordered Fields

- (where )