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# 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?

1. Get an understanding for the statement by looking into example
2. 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:

\begin{align*}\mathbb{R} &\subseteq \mathbb{Z}=\{0,1,-2,2,-2,\ldots\}\\ &\subseteq \mathbb{Q} = \{\frac{m}{n}: m\in \mathbb{Z}, n \in \mathbb{Z}, n\neq 0 \} \\ &\subseteq \mathbb{N=\{1,2,3,\ldots\}}\end{align*}

If $n$ is an integer, we might use the symbol $n \in \mathbb{Z}$

In $\mathbb{R}$

We have an ‘equality’ “=”

This equality has the following properties:

• $x=x$ for all $x \in \mathbb{R}$ (reflective)
• If $x= y$ then $y=x$ (symmetric)
• If $x=y$ and $y=z$ then $x=z$ (transitive)

We have operations:
$+ \\ \times$

​For us: we can add multiply specific number as we used to: $3\cdot 4 = 12\\5+2=7$ do not (yet) use it with symbols $a+a=2a$

We have ordering: $\geq,>\\\leq,<$ don't worry about cardinality yet

# Lec 3

Example 1.2

a)

If $a,b,c$ are integers, and $c=a+b$, then $a=c-b$

Sol'n in book

b)

If $a,b,c,m,n$ are integers and if $am=b$ and $bn=c$, then $c=amn$

c)

If $a,b,c$ are integers, and $a+b=c$, then $-a-b=-c$

### 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 $n$ be an integer, we define the absolute value of $n$, denoted by $|n|$, by

Example, $|100| = 100, |-79| = 79$

#### Divisibility

Suppose, $a,b$ are integers, we say $a$ divides $b$ or that $b$ is divisible by $a$, denoted by $a\mid b$

If there exists an integer $m$ such that $am=b$

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

and we denote that by $a\not \mid b$

Example: $56 = 7 \cdot 8$, so $7 \mid 56$

$13 \not \mid 1$: For any integer m, with $m \cdot 13 = 1$ by EPI 2, $m=1$ and $13 = 1$, or $m=-1$ and $13 = -1$, either is not true