Summary: 1. Sets, relations and functions.
1.1. Set theory. We assume the reader is familiar with elementary set theory as
it is used in mathematics today. Nonetheless, we shall now give a careful treatment
of set theory if only to to allow the reader to become conversant with our notation.
Our treatment will be naive and not axiomatic. For an axiomatic treatment of set
theory we suggest that the reader consult the Appendix to General Topology by
J.L. Kelley where one will find a concise and elegant treatment of this subject as
well as other references for this subject.
By an object we shall mean any thing or entity, concrete or abstract, that might
be a part of our discourse. A set is a collection of objects and is itself an object.
Whenever A is a set and a is one of the objects in the collection A we shall write
and say a is a member of A. A set is determined by its members; that is, if A
and B are sets then
(1) A = B if and only if for every x, x A x B;
this is an axiom; in other words, it is an assumption we make.
The most common way of defining sets is as follows. Suppose P(x) is a formula
in the variable x. We will not go into just what this might mean other than to say
that (i) if y is a variable then P(y) is a formula and (ii) if a is an object and if each
occurrence of x in P(x) is replaced by a then the result P(a) is a statement; we