0.1 The Language of Set Theory

It is assumed that the reader is familiar with the basic concepts of set theory; the following discussion is meant mainly to fix our terminology.
Our notation for the fundamental number systems is as follows.

$$\begin{align} \mathbb{N} &= \text{the set of positive integers not including zero} \\ \mathbb{Z} &= \text{the set of integers} \\ \mathbb{Q} &= \text{the set of rational numbers} \\ \mathbb{R} &= \text{the set of real numbers} \\ \mathbb{C} &= \text{the set of complex numbers} \end{align}$$

Our notation for set logic is as follows.

$$\begin{align} \emptyset &= \text{the empty set} \\ \mathcal{P}(X) &= \text{the power set of set } X \end{align}$$ Given collection of sets $\mathcal{E}$, we have $$\begin{align} \bigcup_{E \in \mathcal{E}} E &= \left\{ x: x \in E \text{ for some } E \in \mathcal{E} \right\} \\ \bigcap_{E \in \mathcal{E}} E &= \left\{ x: x \in E \text{ for all } E \in \mathcal{E} \right\} \end{align}$$ Given indexed family of sets $\left\{ X_{a} \right\}_{a \in A}$, we have $$\begin{align} \bigcup_{a \in A} X_{a} &= \left\{ x: x \in X_{a} \text{ for some } a \in A \right\} \\ \bigcap_{a \in A} X_{a} &= \left\{ x: x \in X_{a} \text{ for all } a \in A \right\} \end{align}$$ A family of sets $\mathcal{E} = \left\{ E_{a} \right\}_{a \in A}$ is called disjoint if $E_{a} \cap E_{b} = \emptyset$ whenever $a \neq b$.

When the index set is $\mathbb{N}$, then we just denote $\left\{ E_{n} \right\}_{n \in \mathbb{N}}$ as $\left\{ E_{n} \right\}_{1}^{\infty}$. In this situation, we define $$\begin{align} \limsup E_{n} &= \cap_{n=1}^{\infty} \cup_{i=n}^{\infty} E_{n} \\ \liminf E_{n} &= \cup_{n=1}^{\infty} \cap_{i=n}^{\infty} E_{n} \end{align}$$ If $E$ and $F$ are sets, we denote their difference by $E \setminus F$: $$E \setminus F = \left\{ x: x \in E \text{ and } x \not\in F \right\}$$ and their symmetric difference $$E \triangle F = (E \setminus F) \cup (F \setminus E)$$ When $E$ is subset of some universal set $X$, we denote $$E^{c} = X \setminus E$$ In this situation we have deMorgan's Laws: $$\begin{align} \left( \bigcup_{a \in A} E_{a} \right)^{c} &= \bigcap_{a \in A} E_{a}^{c} \\ \left( \bigcap_{a \in A} E_{a} \right)^{c} &= \bigcup_{a \in A} E_{a}^{c} \end{align}$$

Our notation for relations and functions is as follows.

If $X$ and $Y$ are sets, we denote their Cartesian product $$X \times Y = \left\{ (x, y): x \in X \text{ and } y \in Y \right\}$$ A relation from $X$ to $Y$ is a subset $R$ of $X \times Y$. If $X = Y$, then we just call it relation on $X$. Important relations are follows.

An equivalence relation on $X$ is a relation $R$ on $X$ having following properties. $$\begin{align} \text{Reflexive} &: (x, x) \in R \text{ for all } x \in X \\ \text{Symmetric} &: \text{if } (x,y) \in R, \text{ then } (y, x) \in R \\ \text{Transitive} &: \text{if } (x,y) \in R \text{ and } (y, z) \in R, \text{ then } (x, z) \in R \end{align}$$

A function $f: X \rightarrow Y$ is a relation $R$ from $X$ to $Y$ with the property that for every $x \in X$, there is a unique $y \in Y$ such that $(x, y) \in R$, in which case we write $f(x) = y$.

If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are mappings, we define function composition $$g \circ f: X \rightarrow Z,\ (g \circ f)(x) = g(f(x))$$

For $f: X \rightarrow Y$, $D \subset X$ and $E \subset Y$, we denote $$\begin{align} f(D) &= \left\{ y \in Y: f(x) = y \text{ for some } x \in D \right\} \text{ (image of } D)\\ f^{-1}(E) &= \left\{ x \in X: f(x) \in E \right\} \text{ (inverse image of } E) \end{align}$$ Now, for the inverse image, following holds. For $\left\{ E_{a} \right\}_{a \in A}$ where $E_{a} \subset Y$, $$\begin{align} f^{-1} \left( \bigcup_{a \in A} E_{a} \right) &= \bigcup_{a \in A} f^{-1}(E_{a}) \\ f^{-1} \left( \bigcap_{a \in A} E_{a} \right) &= \bigcap_{a \in A} f^{-1}(E_{a}) \\ \end{align}$$

For types of functions,

$f: X \rightarrow Y$ is said to be $$\begin{align} \text{injective} &: f(x_{1}) = f(x_{2}) \text{ only when } x_{1} = x_{2} \\ \text{surjective} &: f(X) = Y \\ \text{bijective} &: \text{ injective and surjective } \end{align}$$

Lastly cartesian product for infinite index set.

For indexed family of set $\left\{ E_{a} \right\}_{a \in A}$ $$\prod_{a \in A} E_{a}$$ is the set of functions $f: A \rightarrow \cup_{a \in A} E_{a}$ which satisfies $f(a) \in E_{a}$. If $A = \mathbb{N}$, then we call these functions as sequences.