# Rational Mathematics

## Sequences

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.

### Arrays

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.

### Relationships

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.