The Mathematics of ECE

A rational number $x\in \mathbb{Q}$ is one such that there exists two integers $p,q\in \mathbb{Z}$ for which $x=\frac{p}{q}$.

An irrational number $x\in \mathbb{R}\setminus \mathbb{Q}$ is one which is not rational.

Let $\mathcal{A},\mathcal{B},\mathcal{C}$ be sets. The union is denoted $\cup$ and the intersection $\cap$ with definitions $$\begin{array}{rl}\mathcal{A}\cup \mathcal{B}& =\{x|x\in \mathcal{A}\vee x\in \mathcal{B}\}\\ \mathcal{A}\cap \mathcal{B}& =\{x|x\in \mathcal{A}\wedge x\in \mathcal{B}\}\end{array}$$ which obey the following properties

• $\mathcal{A}\cup \mathcal{B}=\mathcal{B}\cup \mathcal{A}$ and $\mathcal{A}\cap \mathcal{B}=\mathcal{B}\cap \mathcal{A}$ (commutativity)
• $(\mathcal{A}\cup \mathcal{B})\cup \mathcal{C}=\mathcal{A}\cup (\mathcal{B}\cup \mathcal{C})$ and $(\mathcal{A}\cap \mathcal{B})\cap \mathcal{C}=\mathcal{A}\cap (\mathcal{B}\cap \mathcal{C})$ (associativity)
• $\mathcal{A}\cup (\mathcal{B}\cap \mathcal{C})=(\mathcal{A}\cup \mathcal{B})\cap (\mathcal{A}\cup \mathcal{C})$ and $\mathcal{A}\cap (\mathcal{B}\cup \mathcal{C})=(\mathcal{A}\cap \mathcal{B})\cup (\mathcal{A}\cap \mathcal{C})$ (distributivity)

Let $\mathcal{A}$ be some set. If for some $n\in \mathbb{N}$ there exists a one-to-one mapping between elements of $\mathcal{A}$ and $1,2,\dots ,n$, then $\mathcal{A}$ is finite; otherwise, $\mathcal{A}$ is infinite.

Let $\mathcal{A}$ be some infinite set. If there exists a one-to-one mapping between elements of $\mathcal{A}$ and $\mathbb{N}$, then $\mathcal{A}$ is countable; otherwise, $\mathcal{A}$ is uncountable.

For any vector $x\in {\mathbb{R}}^n$ and real number $r\in \mathbb{R}$, we define the (open) ball ${\mathcal{B}}_r(x)$ centered at $x$ with radius $r>0$ as $${\mathcal{B}}_r(x)=\{y\in {\mathbb{R}}^n|||x-y||<r\}$$ We call a subset $\mathcal{A}\subseteq {\mathbb{R}}^n$ an open set if it is some union (finite or infinite) of balls. The empty set is considered open.

A set $\mathcal{A}$ is closed if its complement is an open set.

Let $\mathcal{A}$ be an arbitrary set. A point $x\in \mathcal{A}$ is an interior point of $\mathcal{A}$ if there exists some $r>0$ such that ${\mathcal{B}}_r(x)\subseteq \mathcal{A}$. The collection of all interior points is called the interior of $\mathcal{A}$ and denoted $\mathring{\mathcal{A}}$.

Let $\mathcal{A}$ be an arbitrary set. A point $x\in \mathcal{A}$ is a closure point of $\mathcal{A}$ if for all $r>0$ there exists some $y\in \mathcal{A}$ such that $y\in {\mathcal{B}}_r(x)$. The collection of all closure points is called the closure of $\mathcal{A}$ and denoted $\overline{\mathcal{A}}$.

A neighborhood ${\mathcal{N}}_x$ of a point $x$ is a set such that $x$ is an interior point of ${\mathcal{N}}_x$.