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 ⤴