Set
المؤلف:
Courant, R. and Robbins, H.
المصدر:
"The Algebra of Sets." Supplement to Ch. 2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.
الجزء والصفحة:
...
17-1-2022
2853
Set
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). Members of a set are often referred to as elements and the notation 
is used to denote that 
is an element of a set 
. The study of sets and their properties is the object of set theory.
Older words for set include aggregate and set class. Russell also uses the unfortunate term manifold to refer to a set.
Historically, a single horizontal overbar was used to denote a set stripped of any structure besides order, and hence to represent the order type of the set. A double overbar indicated stripping the order from the set and hence represented the cardinal number of the set. This practice was begun by set theory founder Georg Cantor.
Symbols used to operate on sets include 
(which means "and" or intersection), and 
(which means "or" or union). The symbol 
is used to denote the set containing no elements, called the empty set.
There are a number of different notations related to the theory of sets. In the case of a finite set of elements, one often writes the collection inside curly braces, e.g.,
![A=<span style=]() {1,2,3} " src="https://mathworld.wolfram.com/images/equations/Set/NumberedEquation1.svg" style="height:20px; width:87px" /> |
(1)
|
for the set of natural numbers less than or equal to three. Similar notation can be used for infinite sets provided that ellipses are used to signify infiniteness, e.g.,
![B=<span style=]() {3,4,5,...} " src="https://mathworld.wolfram.com/images/equations/Set/NumberedEquation2.svg" style="height:20px; width:112px" /> |
(2)
|
for the collection of natural numbers greater than or equal to three, or
![C=<span style=]() {...,-4,-2,0,2,4,...} " src="https://mathworld.wolfram.com/images/equations/Set/NumberedEquation3.svg" style="height:20px; width:196px" /> |
(3)
|
for the set of all even numbers.
In addition to the above notation, one can use so-called set builder notation to express sets and elements thereof. The general format for set builder notation is
![<span style=]() {x:p(x)}, " src="https://mathworld.wolfram.com/images/equations/Set/NumberedEquation4.svg" style="height:21px; width:64px" /> |
(4)
|
where 
denotes an element and 
denotes a property 
satisfied by 
. () can also be expanded so as to indicate construction of a set which is a subset of some ambient set 
, e.g.,
![<span style=]() {x in X:p(x)}. " src="https://mathworld.wolfram.com/images/equations/Set/NumberedEquation5.svg" style="height:21px; width:97px" /> |
(5)
|
It is worth noting is that the ":" in () and () is sometimes replaced by a vertical line, e.g.,
![<span style=]() {x in X|p(x)}. " src="https://mathworld.wolfram.com/images/equations/Set/NumberedEquation6.svg" style="height:22px; width:98px" /> |
(6)
|
Also worth noting is that the sets in (), (), and () can all be rewritten in set builder notation as subsets of the set 
of integers, namely
respectively.
Other common notations related to set theory include 
, which is used to denote the set of maps from 
to 
where 
and 
are arbitrary sets. For example, an element of 
would be a map from the natural numbers 
to the set 
. Call such a function 
, then 
, 
, etc., are elements of 
, so call them 
, 
, etc. This now looks like a sequence of elements of 
, so sequences are really just functions from 
to 
. This notation is standard in mathematics and is frequently used in symbolic dynamics to denote sequence spaces.
Let 
, 
, and 
be sets. Then operation on these sets using the 
and 
operators is commutative
 |
(10)
|
 |
(11)
|
associative
 |
(12)
|
 |
(13)
|
and distributive
 |
(14)
|
 |
(15)
|
More generally, we have the infinite distributive laws
 |
(16)
|
 |
(17)
|
where 
runs through any index set 
. The proofs follow trivially from the definitions of union and intersection.
REFERENCES
Courant, R. and Robbins, H. "The Algebra of Sets." Supplement to Ch. 2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.
Oxford, England: Oxford University Press, pp. 108-116, 1996.
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