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Posted 28 February 2008

SETS: NOTATION

This section describes customary usage for defining specific sets.   Notations for various constructions and properties of sets is discussed under the appropriate topic.

Contents

Sets of numbers. 1

Element notation. 1

List notation. 1

Setbuilder notation. 2

Variations on setbuilder notation. 3

Appendices. 4

Sets of numbers

The following notation for sets of numbers is fairly standard.  

¨   is the set of all natural numbers.

¨   is the set of all integers.  [Why?]

¨   is the set of all rational numbers.

¨   is the set of all real numbers.

Some authors use  for ,  but that symbol is also used for the unit interval.  Many authors use  to denote the nonnegative integers instead of the positive ones.       

Element notation

If x is a member of the set A, you may write "  ".  This is pronounced in any of the following ways:

¨  " x is in S".

¨  " x is an element of  S".

¨  " x is a member of  S".

¨  " S contains x" or " x is contained in S".   This can be ambiguous. 

If x  is not a member of A, you may write "  ".

Examples

, , but .  However,  and .

List notation

There are two common methods for defining sets: list notation, discussed here, and setbuilder notation.

Definition: list notation

A set with a small number of members may be denoted by listing them inside curly brackets.  The list must include exactly all of the elements of the set and nothing else. 

Example

The set  contains the numbers 1, 3 and  as elements, and no others. So  but .

Properties of list notation

If a occurs in a list notation then a is in the set the notation defines.  If it does not occur, then it is not in the set.  (See the fine point about this statement.)  Consequences:

¨  The order in which the elements are given is irrelevant: {2,5,6} and {5,2,6} are the same set. 

¨  Repetitions don't matter, either [why?]: {2,5,6}, {2,2,5,6} and {2,5,5,5,6,6} are all the same set.  The set {2,5,5,6,6} has three elements. 

¨  The symbols ‘{2,5,6}’, ‘{2,2,5,6}’ and ‘{2,5,5,5,6,6}’  are different representations of the same set.

See also the discussion under comprehension.

Sets are arbitrary

Examples

¨   is a set.   There is no point in asking, “Why did you put that 6 in there?!”  (Sets can be arbitrary.)

¨  Let f  be the function on the reals for which .  Let M be the matrix .  Then  is a set.  Sets do not have to be homogeneous in any sense.

Setbuilder notation

Definition: setbuilder notation

A set may be denoted by the expression , where P is an assertion. This denotes the set of all elements for which the assertion P(x) is true.  The set  contains no other elements.

¨  The notation “  ” is called setbuilder notation.

¨  The assertion P is called the defining condition for the set. 

¨  The set  is called the truth set of the assertion P. 

¨  Examples

¨  The notation  denotes the  set {2, 3, 4, 5}. The defining condition is “n is an integer and  ” .  The set {2, 3, 4, 5} is the truth set of the assertion “n is an integer and  ”.

¨  The notation  denotes the set .

¨  If x is a real variable, the notation  denotes the infinite set of all real numbers bigger than 6.  For example,  and .

¨  The set  defined by  has among its elements 0, 1/4, , 1, and an infinite number of other numbers.  is fairly standard notation for this set  it is called the unit interval.

Problems with setbuilder notation

¨  Some aspects of setbuilder notation are irrelevant.   The assertion P can be worded in different ways.  For example, . 

¨