One of the principal properties of language, and of argument, is that the letters,
words, sentences, paragraphs and so on appear one after another. We will call any such
arrangement a **sequence** (or *series* or *succession*).
The symbols appearing in the sequence are its **terms**. Any sequence has a **first**
and a **last** term. Each term in a sequence except the last has a uniquely defined
**successor** which comes just after it, and each term except the first has a uniquely
defined **predecessor** which comes just before it. [Note: the concept of a sequence
that 'goes on for ever' with no last term is discussed in the section on Infinity.]

There can be duplicated terms in a sequence. A sequence in which all the terms are
different we call a **list**. A sequence in which all the terms are the same we call
a **tally**. A tally using the symbol | (chosen because it takes up little space) we
will call a **standard tally**. The standard tally derived from a sequence by replacing
each of its terms by | we call its **length**.

A sequence can be regarded as an entity in its own right and denoted by a single symbol. Thus we can write s = (a, b, ..., k) using rounded brackets to enclose the terms of s. It follows that if s = (a, b, ..., k) and t = (m, n, ..., w) then by equality of these sequences, written s = t, we mean that a = m, b = n, and so on, up to k = w. It is clear that if s = t then the sequences s and t must have the same length.

For some purposes it is convenient to suppose that we have a **standard order** for
all the symbols that we use. We will take this as being the **alphanumerical order**,
that is alphabetical order of letters, followed by numerical order of numerals, and other
symbols following after those. (But any other order could be designated as the standard.)
A **standard sequence** is then a sequence whose terms are in standard order.
A standard list is thus a representation of a set.

An **array** is a sequence of sequences. For purposes of visualisation an
array is often presented in two-dimensional form with the terms of the component
sequences being written across the page and the sequences themselves listed down
the page. For example the array ((A, B, C, D), (E, F, G), (H, I), (J)) can be
represented as:

(A, B, C, D)

(E, F, G)

(H, I)

(J)

The terms of the terms of an array are called its **elements**.

A **table** (or *rectangular array*) is an array in which all the
component sequences are of the same length. The component sequences of a table
are called its **rows**.

A proposition R involving one or more variables, which is true or false when the
variables are given any of their values, we call a **relationship**,
which can be symbolised as R(x, y, z, ...).

A relationship R without a variable reduces to a proposition.

A relationship R(x) involving one variable we call a **condition**.

A relationship R(x, y) involving two variables we call a **relation**.

The **cartesian product** of sets A, B, C, ... is the set of all sequences
(a, b, c, ...) with a in A, b in B, c in C and so on, and we denote it by ×(A, B, C, ...).
We also write ×(A, B) as A×B. For any range A we have O×A = A×O = O. A relationship
expressed by a statement s(x, y, z. ..) may be identified with the range of sequences
of values (a, b, c, ...) that the variables (x, y, z, ...) can take that make the
relationship true.