Note 1  Introduction
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:
 A set $S$ consists of elements, and if $a$ is one of these elements, we write $a \in S$.
 There exists exactly one set with no elements.
 The setse can be described by either giving characterizing property, or by listing all elements.
 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

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