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Posted 6 February 2016
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.
 N is the set of all natural numbers.
 Z is the set of all integers. [Why?]
  is the set of all rational numbers.
 C is the set of all real numbers.
Some authors use I for Z , but that symbol is also used for the unit interval. Many authors use N
to denote the nonnegative integers instead of the positive ones.
Element notation
If x is a member of the set A, you may write " x Î A ". 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 " x Ï A ".
Examples
4 Î Z , - 5 Î Z , but
4
4
4
Ï Z . However, Î Q and Î R .
3
3
3
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 {1,3, p } contains the numbers 1, 3 and p as elements, and no others. So 3 Î {1,3, p } but
- 3 Ï {1,3, p } .
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
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
 {1,3,5,6,7,9,11} is a set. There is no point in asking, “Why did you put that 6 in there?!” (Sets
can be arbitrary.)
æ2 - p ö
÷
÷
 Let f be the function on the reals for which f ( x) = x 3 - 2 . Let M be the matrix çç
÷. Then
èç7
÷
0 ø
ìï 2
1ü
ï
í p , M , f , 42,- 2 ý is a set. Sets do not have to be homogeneous in any sense.
ïîï
e ïþ
ï
Setbuilder notation
Definition: setbuilder notation
{x | P (x)}, where P is an assertion. This denotes the set
of all elements for which the assertion P(x) is true. The set {x | P ( x)} contains no other elements.
 The notation “ {x | P ( x)} ” is called setbuilder notation.
A set may be denoted by the expression
 The assertion P is called the defining condition for the set.
 The set {x | P ( x)} is called the truth set of the assertion P.
Examples
 The notation {n | n is an integer and 1< n < 6} denotes the set {2, 3, 4, 5}. The defining condition is “n is an integer and 1 < n < 6 ” . The set {2, 3, 4, 5} is the truth set of the assertion “n is an integer and 1 < n < 6 ”.
{
}
 The notation x | x 2 - 4 = 0 denotes the set {2,- 2} .
 If x is a real variable, the notation
{x | x > 6}denotes the infinite set of all real numbers bigger
than 6. For example, 6 Ï {x | x > 6} and 17p Î {x | x > 6}.
 The set I defined by I = {x | x is real and 0 £ x £ 1} has among its elements 0, 1/4, p / 4 , 1,
and an infinite number of other numbers. I 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, {x | x > 6}= {x | x ³ 6 and x ¹ 6}.
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
{x | x > 6}could be written
{x : x > 6}.
 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 1 < n < 6 ” 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
2
example, the solution set of the equation x - 4 = 0 is {2,- 2}.
 “ {n | n is an integer and 1< n < 6}” 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 I could be defined as
I = {x Î R | 0 £ x £ 1}
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:
I = {x | x Î R and 0 £ x £ 1}
Other expressions on the left side
Other kinds of expressions occur before the vertical line in setbuilder notation as well.
Example
{
} consists of all the squares of integers; in other words its elements are
This definition could be rewritten as {m | there is an n Î Z such that m = n2 }.
The set n2 | n Î Z
0,1,4,9,16,….
Example
Let A = {1,3,6}. Then {n - 2 | n Î A}= {1,- 1,4}. [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
{n
2
| n Î Z and n ¹ 3 , It is true that 9 = 32 and that 3 is excluded by the defining condition, but it is
}
also true that 9 = (- 3)2 and -3 is not an integer ruled out by the defining condition.
Example
R ¹
{x2 | x Î R} since squares of real numbers can’t be negative.
But R =
{x3 | x Î R}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 {{1,2,3}, {3,4},3,4}. The elements listed include the set {1, 2, 3} among others, but not the number 2. The set
{{1,2,3}, {3,4},3,4}
contains four elements, two sets and two numbers.
Another way of saying this is that the element relation is not transitive: The facts that A Î B and
B Î C do not imply that A Î C . Back
Answers to Questions
Why do we use Z to denote the integers? 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 I to denote the
integers except that it is very common to use it to denote the real numbers between 0 and 1 inclusive.
Q for the rationals presumably comes from “quotient”.
In any case if you are going to read abstract math you will repeatedly come across Z and Q , 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 {n - 2 | n Î A} = {- 1, 1, 4} 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 {- 1,1,4} .