Sets and functionsSet Builder Notation
It's often useful to define a set in terms of the properties its elements are supposed to have.
If is a set and is a property which each element of either satisfies or does not satisfy, then
denotes the set of all elements in which have the property . This is called set builder notation. The colon is read as "such that".
Suppose the set denotes the set of all real numbers between 0 and 1. Then can be expressed as
Counting the number of elements in a set is also an important operation:
Given a set , the cardinality of , denoted , denotes the number of elements in .
Let . Then =
Solution. There are three values with the property that . Therefore, .
Not every set has an integer number of elements. Some sets have more elements than any set of the form . These sets are said to be
Definition (Countably infinite)
A set is countably infinite if its elements can be arranged in a sequence.
The set is
The set of rational numbers between 0 and 1 is countably infinite, since they all appear in the sequence
The set of all real numbers between 0 and 1 is
Show that the set of all ordered pairs of positive integers is countably infinite.
Solution. We visualize the pairs as a grid of points in the first quadrant. We arrange the points in a sequence by beginning in the lower left corner at and snake through the grid: we go right to , then diagonally northwest to , then up to , then diagonally southeast to and through to , and so on.