TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, two sets are said to be disjoint sets if they have no
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
in common. Equivalently, two disjoint sets are sets whose
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
is the
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of more than two sets is called disjoint if any two distinct sets of the collection are disjoint.

# Generalizations

This definition of disjoint sets can be extended to a
family of setsIn set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, s ...
$\left\left(A_i\right\right)_$: the family is pairwise disjoint, or mutually disjoint if $A_i \cap A_j = \varnothing$ whenever $i \neq j$. Alternatively, some authors use the term disjoint to refer to this notion as well. For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if $A_i \cap A_j = \varnothing$ whenever $A_i \neq A_j$ (every two ''distinct'' sets in the family are disjoint).. For example, the collection of sets is disjoint, as is the set of the two parity classes of integers; the family $\left(\\right)_$ with 10 members is not disjoint (because the classes of even and odd numbers are each present five times), but it is pairwise disjoint according to this definition (since one only gets a non-empty intersection of two members when the two are the same class). Two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
s whose intersection is a
finite set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
may be said to be almost disjoint. In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, there are various notions of
separated sets In topology and related branches of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint closures or disjoint
neighborhoods A neighbourhood (British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval E ...
. Similarly, in a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
,
positively separated setsIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
are sets separated by a nonzero
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...
.

# Intersections

Disjointness of two sets, or of a family of sets, may be expressed in terms of intersections of pairs of them. Two sets ''A'' and ''B'' are disjoint if and only if their intersection $A\cap B$ is the
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

. It follows from this definition that every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself. If a collection contains at least two sets, the condition that the collection is disjoint implies that the intersection of the whole collection is empty. However, a collection of sets may have an empty intersection without being disjoint. Additionally, while a collection of less than two sets is trivially disjoint, as there are no pairs to compare, the intersection of a collection of one set is equal to that set, which may be non-empty. For instance, the three sets have an empty intersection but are not disjoint. In fact, there are no two disjoint sets in this collection. Also the empty family of sets is pairwise disjoint. A
Helly familyIn combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to ...
is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the
closed interval In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s form a Helly family: if a family of closed intervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it must be pairwise disjoint.

# Disjoint unions and partitions

A
partition of a set The traditional Japanese symbols for the 54 chapters of the '' Tale of Genji'' are based on the 52 ways of partitioning five elements (the two red symbols represent the same partition, and the green symbol is added for reaching 54). In mathemati ...

''X'' is any collection of mutually disjoint non-empty sets whose union is ''X''., p. 28. Every partition can equivalently be described by an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
that describes whether two elements belong to the same set in the partition.
Disjoint-set data structure In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Alg ...
s and
partition refinementIn the design of algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the ...
are two techniques in computer science for efficiently maintaining partitions of a set subject to, respectively, union operations that merge two sets or refinement operations that split one set into two. A
disjoint union In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

may mean one of two things. Most simply, it may mean the union of sets that are disjoint. But if two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make them disjoint before forming the union of the modified sets. For instance two sets may be made disjoint by replacing each element by an ordered pair of the element and a binary value indicating whether it belongs to the first or second set. For families of more than two sets, one may similarly replace each element by an ordered pair of the element and the index of the set that contains it..

# See also

*
Hyperplane separation theorem In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
for disjoint convex sets *
Mutually exclusive events In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument ...
*
Relatively prime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
, numbers with disjoint sets of prime divisors * Separoid * Set packing, the problem of finding the largest disjoint subfamily of a family of sets

# External links

* {{DEFAULTSORT:Disjoint Sets
Basic concepts in set theory{{Commons This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed. Mathematical concepts ...
Set families
The provided text passage is an excerpt from a mathematical article or textbook, discussing the concept of disjoint sets, their properties, and related mathematical ideas. It does not contain any specific mathematical questions or answers to such questions.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, two sets are said to be disjoint sets if they have no
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
in common. Equivalently, two disjoint sets are sets whose
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
is the
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of more than two sets is called disjoint if any two distinct sets of the collection are disjoint.

# Generalizations

This definition of disjoint sets can be extended to a
family of setsIn set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, s ...
$\left\left(A_i\right\right)_$: the family is pairwise disjoint, or mutually disjoint if $A_i \cap A_j = \varnothing$ whenever $i \neq j$. Alternatively, some authors use the term disjoint to refer to this notion as well. For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if $A_i \cap A_j = \varnothing$ whenever $A_i \neq A_j$ (every two ''distinct'' sets in the family are disjoint).. For example, the collection of sets is disjoint, as is the set of the two parity classes of integers; the family $\left(\\right)_$ with 10 members is not disjoint (because the classes of even and odd numbers are each present five times), but it is pairwise disjoint according to this definition (since one only gets a non-empty intersection of two members when the two are the same class). Two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
s whose intersection is a
finite set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
may be said to be almost disjoint. In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, there are various notions of
separated sets In topology and related branches of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint closures or disjoint
neighborhoods A neighbourhood (British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval E ...
. Similarly, in a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
,
positively separated setsIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
are sets separated by a nonzero
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...
.

# Intersections

Disjointness of two sets, or of a family of sets, may be expressed in terms of intersections of pairs of them. Two sets ''A'' and ''B'' are disjoint if and only if their intersection $A\cap B$ is the
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

. It follows from this definition that every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself. If a collection contains at least two sets, the condition that the collection is disjoint implies that the intersection of the whole collection is empty. However, a collection of sets may have an empty intersection without being disjoint. Additionally, while a collection of less than two sets is trivially disjoint, as there are no pairs to compare, the intersection of a collection of one set is equal to that set, which may be non-empty. For instance, the three sets have an empty intersection but are not disjoint. In fact, there are no two disjoint sets in this collection. Also the empty family of sets is pairwise disjoint. A
Helly familyIn combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to ...
is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the
closed interval In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s form a Helly family: if a family of closed intervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it must be pairwise disjoint.

# Disjoint unions and partitions

A
partition of a set The traditional Japanese symbols for the 54 chapters of the '' Tale of Genji'' are based on the 52 ways of partitioning five elements (the two red symbols represent the same partition, and the green symbol is added for reaching 54). In mathemati ...

''X'' is any collection of mutually disjoint non-empty sets whose union is ''X''., p. 28. Every partition can equivalently be described by an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
that describes whether two elements belong to the same set in the partition.
Disjoint-set data structure In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Alg ...
s and
partition refinementIn the design of algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the ...
are two techniques in computer science for efficiently maintaining partitions of a set subject to, respectively, union operations that merge two sets or refinement operations that split one set into two. A
disjoint union In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

may mean one of two things. Most simply, it may mean the union of sets that are disjoint. But if two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make them disjoint before forming the union of the modified sets. For instance two sets may be made disjoint by replacing each element by an ordered pair of the element and a binary value indicating whether it belongs to the first or second set. For families of more than two sets, one may similarly replace each element by an ordered pair of the element and the index of the set that contains it..

# See also

*
Hyperplane separation theorem In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
for disjoint convex sets *
Mutually exclusive events In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument ...
*
Relatively prime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
, numbers with disjoint sets of prime divisors * Separoid * Set packing, the problem of finding the largest disjoint subfamily of a family of sets

# External links

* {{DEFAULTSORT:Disjoint Sets
Basic concepts in set theory{{Commons This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed. Mathematical concepts ...
Set families
user
The provided text passage is an excerpt from a mathematical article or textbook, discussing the concept of disjoint sets, their properties, and related mathematical ideas. It does not contain any specific mathematical questions or answers to such questions.

TYPE: NO_QUESTION