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Posted 28 February 2008
This section describes customary usage for defining specific sets. Notations for various constructions and properties of sets is discussed under the appropriate topic.
Variations on
setbuilder notation
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.
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 " ".
,
,
but
. However,
and
.
There are two common methods for defining sets: list notation, discussed here, and setbuilder 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.
The set contains the numbers
as elements, and no others. So
but
.
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.
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.
¨
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.
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.
¨
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.
¨