## Axioms and Primitive Concepts

• axioms and primitive concepts are defined
• Then, we say that statement is true if it follows from axioms and primitive statements; false if converse follows

## Sets and Equivalence Relations

Axioms of sets:

1. A set $S$ consists of elements, and if $a$ is one of these elements, we write $a \in S$.
2. There exists exactly one set with no elements.
3. The setse can be described by either giving characterizing property, or by listing all elements.
4. For every element $a$, exactly one of the following is true:
• $a \in S$
• $a \not \in S$

Definition 1: Subset

A set $B$ is a subset of a set $A$ if every element of $B$ is an element of $A$. In formal language. we can write

$\forall x \in B \implies x \in A$

In this case we write $$B \subseteq A$$.1

Definition 2: Cartesian Product

Let $A$, $B$ be sets. Then,

$A \cross B = \{ (a, b) | a \in A , b \in B\}$

is called a Cartesian Product of $A$ and $B$.

Relations between sets:

Let $A$, $B$ be sets. We say that $$\mathbb{R}$$ is a relation between $A$ and $B$ if $$\mathbb{R} \subseteq A \cross B$$. For every $$a \in A$$ and $$b \in B$$, we write $$a \mathbb{R} b$$ if and only if $$(a, b) \in \mathbb{R}$$.

Definition 3: relations

We say that a function $F: A \to B$ is a relation $F \subseteq A \cross B$ such that

• $a_1 \mathbb{F} b_1$ and $a_2 \mathbb{F} b_2$ implies $a_1 \neq a_2$.
• For every $a \in A$ there exists $b \in B$ such that $a \mathbb{F} b$.

## Footnotes

1. $\subset$ is strict