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Posted 22 April 2009

# 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.

Sets of numbers

# 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.

### Sets are arbitrary

Any mathematical object can be the element of a set.

The elements of a set do not have to have anything in common.

The elements of a set do not have to form a pattern.

#### 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, .

A set can be expressed in many different ways in setbuilder notation.

¨  Setbuilder notation has another problem which you usually don’t need to worry about; see Russell’s Paradox if you are curious.

### Usage

¨  A colon may be used instead of “|” in the setbuilder notation.  Thus  could be written .

¨  The truth set for an assertion P may be called the extension of P by logicians and some mathematicans.  They would say that the extension of the assertion “n is an integer and  ” is the set {2, 3, 4, 5}.  Mathematicians usually mean something else by “extension”.

¨  When the assertion P is an equation, the truth set of P may be called the solution set of P.  For example, the solution set of the equation  is .

¨   ” may be pronounced “The set of integers such that 1 < n < 6”.  The phrase “The set of integers such that 1 < n < 6” means exactly the set {2, 3, 4, 5}.   See the.

# Variations on setbuilder notation

An expression may be used left of the vertical line in setbuilder notation, instead of a single variable.

### Giving the type of the variable

You can use an expression on the left side of setbuilder notation to indicate the type of the variable.

#### Example

The unit interval  could be defined as

making it clear that it is a set of real numbers rather than, say rational numbers.  You can always get rid of the expression to the left of the vertical line by complicating the defining condition, like this:

### Other expressions on the left side

Other kinds of expressions occur before the vertical line in setbuilder notation as well.

#### Example

The set  consists of all the squares of integers; in other words its elements are 0,1,4,9,16,….  This definition could be rewritten as .

#### Example

Let .  Then .  [Why did you list them that way?]

### Warning

Be careful when you read such expressions. For example, the integer 9 is an element of the set , It is true that  and that 3 is excluded by the defining condition, but it is also true that  and -3 is not an integer ruled out by the defining condition.

#### Example

since squares of real numbers can’t be negative.  But  since every real number has a cube root.

# Appendices

## Fine point about list notation

When (some of) the elements in list notation are themselves sets (more about that here), care is required.  For example, the numbers 1 and 2  are not elements of the set .  The elements listed include the set {1, 2, 3} among others, but not the number 2.  The set  contains four elements, two sets and two numbers.

Another way of saying this is that the element relation is not transitive: The facts that  and  do not imply that .  Back

Answer: The dominant language for math in the nineteenth century, indeed up until Hitler, was German, in which the word for number is “Zahl”.  I would have no problem with using   to denote the integers except that it is very common to use it to denote the real numbers between 0 and 1 inclusive.

for the rationals presumably comes from “quotient”.

In any case if you are going to read abstract math you will repeatedly come across  and , so get over it.

What’s the point of allowing repetition in set notation?  Back

Answer:  It can easily happen that you want to list symbols that give all the elements of a set but you don’t know whether all the symbols denote different mathematical objects.   Need example here. Or it could happen than you have a very large output from a computer calculation and it is difficult to check that all the entries are different.

Why didn’t you say  so the elements would be in the same order as the original?   Back.

Answer:  To make you ask this question!  Order does not matter in list notation.  The answer is not wrong, but it would have been better writing style to give the answer as .