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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.
Contents
Variations on
setbuilder notation
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
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: {
¨
Repetitions
don't matter,
either [why?]:
{
¨ 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
¨
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 {
” . The
set {
”.
¨
The notation denotes the set
.
¨
If x is a real variable, the notation denotes the infinite set of all real numbers
bigger than
and
.
¨
The set defined by
has among its elements
,
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 {
¨
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
.
Example
Let . Then
.
[Why
did you list them that way?]
Warning
Be careful when you read such expressions. For example, the
integer ,
It is true that
and that
and -
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
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 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
.