MATH 109 Lecture 3

Definitions and axioms

Axiom

Statement that we assume to be true

Assuming familiarity with

• Natural numbers (): 0, 1, 2, 3, …
• Integers ()
• Rational numbers ()
• Real numbers ()

Let $a,b$ be two integers

We say divides if there exists an integer s.t.

If $a$ divides $b$

We say that is a multiple of And that is a factor of We can also denote divides as

An integer $m$ is even if $2|m$

An integer $m$ is odd if $m$ is not even

0 is even because

0 is an even number

2 divides 0 By definition of even numbers, 0 is an even number

The sum of two even numbers is even

Let be two even numbers such that their sum is even. There exists two integers such that The sum of two integers is an integer is an integer, therefore 2 divides . Therefore, must be even

1 is not even (1 is odd)

- not a proof because we need to prove that is not an integer Circular reasoning

Axioms of Orders (3.1.2 in the textbook)

• - one of them is true
• If then , and if then
• If then , and if , then
• If then

Let be a natural number
Case 1: If then so
Case 2: If then so
Case 3: If then so
So in all cases, so 1 is odd