    Next: 1.2 Propositions Up: 1. Notation, Undefined Concepts Previous: 1. Notation, Undefined Concepts   Index

# 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  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.

1.1   Definition (Set.) A set is a collection of objects. A small set is often described by listing the objects it contains inside curly brackets, e.g., denotes the set of positive odd integers smaller than ten.

1.2   Notation ( .) A few sets appear so frequently that they have standard names: 1.3   Notation ( .) If is a set and is an object, we write (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 .

1.4   Example. Thus we have            (1.5)

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

1.6   Definition (Subset, .) Let and be sets. We say that is a subset of and write if and only if every object in is also in .

1.7   Example. are all true statements. However (1.8)

is not a statement, but an ungrammatical phrase, since has only been defined when and are sets, and is not a set.

1.9   Definition (Set equality.) Two sets and are considered to be the same if and only if they contain exactly the same objects. In this case we write Thus if and only if and .

1.10   Example.     Next: 1.2 Propositions Up: 1. Notation, Undefined Concepts Previous: 1. Notation, Undefined Concepts   Index