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Posted 5 December 2006

SOME SPECIFIC SETS

Intervals

Definition: interval

Let a and b be real numbers.  We use the notation  for the set .  This is called the open interval from a to b.  The closed interval  from a to b is the set  

Example

The unit interval  is defined to be the interval .

Warning

The notation  is also used to mean the ordered pair with first coordinate a and second coordinate b.

Picturing intervals

You can picture the set of real numbers as a line infinitely long in both directions.  Numbers a and b may be pictured as points on the line, this way:

The open interval  is the set of points between a and b, not including a and b. The closed interval  is the set of points between a and b, including a and b.

The empty set

Definition:empty set

The empty set is the unique set with no elements at all. It is denoted by “{ }” or “  ”.  The existence and uniqueness of the empty set follows directly from the specification for sets.  

Examples

¨  , because every square of a real number is nonegative. 

¨  The interval notation " [a, b]" defines the empty set if a > b. For example, .   That is because by definition , but there are no real numbers that are bigger than 3 and less than 2.

¨  It is also true that . 

¨  Because there is only one empty set, , [3, 2] and (3, 2) are all exactly the same set.  Another way of saying this is that “  ”, “[3, 2]” and “(3, 2)”  are different representations of the same set, namely the empty set.

Empty set as element of a set

Since the empty set is a set, it can be an element of another set. Consider this: although "  " and " { }" both denote the empty set,  is not the empty set; it is a set whose only element is the empty set.   is in fact a singleton set.

Metaphors and images for the empty set

Container

If you think of a set as a container, the empty set is simply an empty container.  The only problem with this is that there is only one empty set, whereas normally you expect that any number of different containers can be empty.

Collection

If you think of a set as a collection, there is a problem:  If I didn’t own any chess pieces, I would not talk about “my collection of chess pieces”.   If you don’t have something, you don’t have a collection of them!

The concept of collection also has the same problem that “container” has:  You would mean two different things by the statements

¨  I don’t have a collection of chess pieces.

¨  I don’t have a collection of Hummel figures.

But there is only one empty set.  It does not come in types!

Pointer

If you think of a set as a pointer to its elements, then the empty set is the same as the null pointer.  Again there is a problem: A computer program can have many different pointers set to null at the same time.  But there is only one empty set. 

As always, images and metaphors for the concept of set produce cognitive dissonance.  Remember, the definition always wins!

Usage  

Singleton sets

Definition:singleton

A set containing exactly one element is called a singleton set.

Examples

¨  {3} is the set whose only element is 3.

¨   is the set whose only element is the empty set.

¨  The closed interval [3, 3] is the singleton set {3}, but the open interval (3, 3) is the empty set.

¨   but  is an infinite set. 

¨   but .

Remark

Because a set is distinct from its elements, a set with exactly one element is not the same thing as the element. Thus {3} is a set, not a number, whereas 3 is a number, not a set.