# Lec 1

Definition: A **statement** is a sentence ( words, mathematical symbols, or a combination of those) that is either true or false. No ambiguity

### Conditional Statement

A statement of the form:

`if ..., then ...`

The if-part is called the **hypothesis** of the statement, the then-part is called the **conclusion** of the statement.

# Lec 2

Given a statement: How do we know what its truth value is?

- Get an understanding for the statement by looking into example
- If the statement is true find a proof

A proof is apiece of writing that demonstrates the statement is true. We often call such statement: Proposition, Lemma, Theorem

We construct proofs using logical arguments and statement which we already know to be true

Statement we are allowed to use and which have not been proven are called axioms. They build the base of our mathematical structure

Calculus:

If is an integer, we might use the symbol

In

We have an **‘equality’** “=”

This equality has the following properties:

- for all (reflective)
- If then (symmetric)
- If and then (transitive)

We have operations:

For us: we can add multiply specific number as we used to: do not (yet) use it with symbols

We have ordering: don't worry about **cardinality** yet

# Lec 3

Example 1.2

a)

If are integers, and , then

Sol'n in book

b)

If are integers and if and , then

Step | Justification |
---|---|

are integers, and | Hypothesis |

Substitution of Equals | |

Transitivity of Equals (and last part of Hypothesis) | |

Symmetry of Equals |

c)

If are integers, and , then

Step | Justification |
---|---|

are integers, | Hypothesis |

exists | Additive Inverses |

Closure | |

Substitution of Equals | |

EPI 4 | |

EPI 4 | |

EPI 7 | |

EPI 6 | |

EPI 7 | |

EPI 6 |

### 1.2 Definitions

A **definition** is an agreement between writer and reader as to the meaning of a word, phase, symbol.

A definition requires no proof.

#### Absolute Value

Definition 1.1

Let be an integer, we define the absolute value of , denoted by , by

Example,

#### Divisibility

Suppose, are integers, we say divides or that is divisible by , denoted by

If there exists an integer such that

If no integer exists such that , we say a does not divide b, or is not divisible by

and we denote that by

Example: , so

: For any integer m, with by EPI 2, and , or and , either is not true