# 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 $x∈V$ ("$x$ 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 $ϕ$.

For example, we may define a vocabulary $V={"and","or"}$ which has cardinality $∣V∣=2$. Then, $"and"∈V$ while $"but"∈V$.

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

- The
**union**$A∪B:={x∣x∈Aorx∈B}$. - The
**intersection**$A∩B:={x∣x∈Aandx∈B}$. - The
**set difference**$A−B:={x∣x∈Aandx∈B}$.

The above uses

set-builder notationto define each set in terms of $A$ and $B$. We read theuniondefinition as "the set of all $x$ such that $x$ is in $A$ or $x$ is in $B$".

**Special sets.** We denote $R$ as the set of all real numbers. For example: $1,1.5,π∈R$. We define $Z$ as the set of all integers and $Z_{+}$ as the set of all positive integers. From these, we define the **natural numbers** $N=Z_{+}∪{0}$.

## 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 $p=(−1,3)$ is an ordered pair.

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

$R={x∣x∈R}.$

Let's generalize this to the set of all points (ordered pairs) in 2D space by using the **Cartesian product $×$ of $R$**: $R_{2}:=R×R:={(x_{1},x_{2})∣x_{1},x_{2}∈R}.$

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 $A$: $A_{k}:=A×⋯×A:={(x_{1},…,x_{k})∣x_{1},…,x_{k}∈A}.$

So the ordered pair $p=(−1,3)∈Z_{2}$ and $(1,2,π,4,5,6)∈R_{6}$.

## 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. $x$, while elements of vectors (scalars) will be lowercase non-boldface, e.g. $a$.

Definition.A $k$-dimensionalvectoris a $k$-tuple with scalar elements: $(x_{1},…,x_{k})∈R_{k}.$

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

Definition.The mathematical operator $+$ is definedelementwisesuch that $x+y:=(x_{1}+y_{1},…,x_{k}+y_{k}).$

Definition.Scalar-vector multiplication is defined as $ax:=(ax_{1},…,ax_{k}).$

Definition.The mathematical operator $−$ is defined elementwise as $x−y:=x+−1⋅y=(x_{1}−y_{1},⋯,x_{k}−y_{k}).$