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3.2 Sets Defined by Propositions

The most common way of describing sets is by means of proposition forms.

3.24   Notation () Let be a proposition form over a set , and let be a subset of . Then
 (3.25)

is defined to be the set of all elements in such that is true. The set described in (3.25) is also written

In cases where the meaning of '' is clear from the context, we may abbreviate (3.25) by

3.26   Examples.

is the set of all even integers, and

If and are sets, then

 (3.27) (3.28) (3.29)

We will use the following notation throughout these notes.

3.30   Notation ( , ) If is an integer we define

Thus

Similarly, if is a real number, we define

3.31   Definition (Ordered pair.) If are objects, then the ordered pair is a new object obtained by combining . Two ordered pairs are equal if and only if Similarly we may consider ordered triples. Two ordered triples are equal if and only if We use the same notation to represent an open interval in and an ordered pair in . The context should always make it clear which meaning is intended.

3.32   Definition (Cartesian product) If are sets then the Cartesian product of and is defined to be the set of all ordered pairs such that
 (3.33)

3.34   Examples. Let be real numbers with and . Then

and

Thus in general .

The set is denoted by . You are familiar with one Cartesian product. The euclidean plane is the Cartesian product of R with itself.

3.35   Exercise. Let and let

Sketch the sets . For you should include an explanation of how you arrived at your answer. For the other sets no explanation is required.

3.36   Exercise. Do there exist sets such that has exactly five elements?

Next: 3.3 Functions Up: 3. Propositions and Functions Previous: 3.1 Propositions   Index
Ray Mayer 2007-09-07