Mathematical Preliminaries

This book is intended to be entirely self-contained. So, we are providing a minimal set of definitions needed to understand the text.

• We indicate a definition by $:=$. This often defines a new notation or operator that cannot be derived from other statements.

Given statements $A$ and $B$:

• $A⟹B$ ("$A$ implies $B$") indicates that $B$ logically follows from $A$ (but not necessarily in the opposite direction).
• $A⟺B$ ("$A$ if and only if $B$") indicates that $A$ follows from $B$ and $B$ follows from $A$ (in both directions).

Sets

A set V is a collection of unique elements. We denote membership by $x\in V$ ("$x$ is an element of $V$" or "$x$ is in $V$"). We denote the size of the set (its cardinality) by $|V|$. The empty set is denoted $\varphi$.

For example, we may define a vocabulary $V=\{\text{"and"},\text{"or"}\}$ which has cardinality $|V|=2$. Then, $\text{"and"}\in V$ while $\text{"but"}\notin V$.

Set operations. Given sets $A$ and $B$, we define:

• The union $A\cup B:=\{x\mid x\in A\text{ or }x\in B\}$.
• The intersection $A\cap B:=\{x\mid x\in A\text{ and }x\in B\}$.
• The set difference $A-B:=\{x\mid x\in A\text{ and }x\notin B\}$.

The above uses set-builder notation to define each set in terms of $A$ and $B$. We read the union definition as "the set of all $x$ such that $x$ is in $A$ or $x$ is in $B$".

Special sets. We denote $\mathbb{R}$ as the set of all real numbers. For example: $1,1.5,\pi \in \mathbb{R}$. We define $\mathbb{Z}$ as the set of all integers and ${\mathbb{Z}}^{+}$ as the set of all positive integers. From these, we define the natural numbers $\mathbb{N}={\mathbb{Z}}^{+}\cup \{0\}$.

Tuples and the Cartesian product

Tuples are often used in conjunction with sets and vectors. An $n$-tuple is an ordered collection of $n$ elements. The $1$-tuple is a singleton, and the $2$-tuple an ordered pair. For example, the point $\mathbf{p}=(-1,3)$ is an ordered pair.

Cartesian product. We can redundantly define the set of all reals as:

$$\mathbb{R}=\{x\mid x\in \mathbb{R}\}.$$

Let's generalize this to the set of all points (ordered pairs) in 2D space by using the Cartesian product $\times$ of $\mathbb{R}$: $${\mathbb{R}}^2:=\mathbb{R}\times \mathbb{R}:=\{(x_1,x_2)\mid x_1,x_2\in \mathbb{R}\}.$$

For any set $A$, we can generally define the $k$-fold Cartesian product of A to produce the set of all $k$-tuples with elements in $A$: $$A^k:=A\times \cdots \times A:=\{(x_1,\dots ,x_k)\mid x_1,\dots ,x_k\in A\}.$$

So the ordered pair $\mathbf{p}=(-1,3)\in {\mathbb{Z}}^2$ and $(1,2,\pi ,4,5,6)\in {\mathbb{R}}^6$.

Vectors

In this book, we will only work with vectors defined on the reals. So, we refer to a real number as a scalar. Vectors will be written in lowercase boldface, e.g. $\mathbf{x}$, while elements of vectors (scalars) will be lowercase non-boldface, e.g. $a$.

Definition. A $k$-dimensional vector is a $k$-tuple with scalar elements: $$(x_1,\dots ,x_k)\in {\mathbb{R}}^k.$$

For any two $k$-dimensional vectors $\mathbf{x}$ and $\mathbf{y}$ and scalar $a$, we define the following:

Definition. The mathematical operator $+$ is defined elementwise such that $$\mathbf{x}+\mathbf{y}:=(x_1+y_1,\dots ,x_k+y_k).$$

Definition. Scalar-vector multiplication is defined as $$a\mathbf{x}:=(a{x}_1,\dots ,a{x}_k).$$

Definition. The mathematical operator $-$ is defined elementwise as $$\mathbf{x}-\mathbf{y}:=\mathbf{x}+-1\cdot \mathbf{y}=(x_1-y_1,\cdots ,x_k-y_k).$$