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 and :

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


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

For example, we may define a vocabulary which has cardinality . Then, while .

Set operations. Given sets and , we define:

  • The union .
  • The intersection .
  • The set difference .

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

Special sets. We denote as the set of all real numbers. For example: . We define as the set of all integers and as the set of all positive integers. From these, we define the natural numbers .

Tuples and the Cartesian product

Tuples are often used in conjunction with sets and vectors. An -tuple is an ordered collection of elements. The -tuple is a singleton, and the -tuple an ordered pair. For example, the point is an ordered pair.

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

Let's generalize this to the set of all points (ordered pairs) in 2D space by using the Cartesian product of :

For any set , we can generally define the -fold Cartesian product of A to produce the set of all -tuples with elements in :

So the ordered pair and .


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

Definition. A -dimensional vector is a -tuple with scalar elements:

For any two -dimensional vectors and and scalar , we define the following:

Definition. The mathematical operator is defined elementwise such that

Definition. Scalar-vector multiplication is defined as

Definition. The mathematical operator is defined elementwise as