1.1 Sets

The ideas discussed in this chapter (e.g. *set*, *proposition*,
*function*) are so basic that I cannot define them in terms of
simpler ideas. Logically they are undefined concepts, even though I
give definitions for them. My ``definitions'' use undefined words
(e.g. *collection*, *statement*, *rule*) that are essentially
equivalent to what I attempt to define. The purpose of these ``definitions''
and examples is to illustrate how the ideas will be used in the later
chapters. I make frequent use of the undefined terms ``true'', ``false'',
and ``there is''.
It might be appropriate to spend some time discussing various
opinions about the meaning of `` truth" and `` existence" in mathematics, but
such a discussion would be more philosophical than mathematical, and would not be very
relevant to anything that follows. If such questions interest you, you might enjoy
reading *Philosophy of Mathematics* by Benacerraf and Putnam [7] or
the article *Schizophrenia in Contemporary Mathematics* by Errett Bishop [10, pp
1-10]

Some of the terms and notation used in the examples in this chapter will be defined more precisely later in the notes. In this chapter I will assume familiar properties of numbers that you have used for many years.

denotes the set of positive odd integers smaller than ten.

(read this as `` is in ") to mean that is an object in , and we write

(read this as `` is not in ") to mean that is not in .

To see why (1.5) is true, observe that the only objects in are , , and . Since

it follows that .

if and only if every object in is also in .

are all true statements. However

(1.8) |

Thus if and only if and .