Equivalence Class 3

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Contents

1 Equivalence class 11.1 Notation and formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Equivalence relation 62.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Well-denedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.10 Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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2.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.11 Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.12 Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.16 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Partition of a set 153.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Renement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Quotient space (linear algebra) 234.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Quotient of a Banach space by a subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4.2 Generalization to locally convex spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Quotient space (topology) 265.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Quotient map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.5 Compatibility with other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.6.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.6.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Set (mathematics) 306.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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6.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.10 De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.15 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.15.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.15.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.15.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Chapter 1

Equivalence class

This article is about equivalency in mathematics. For equivalency in music, see equivalence class (music).In mathematics, when a set has an equivalence relation dened on its elements, there is a natural grouping of

Congruence is an example of an equivalence relation. The two triangles on the left are congruent, while the third and fourth trianglesare not congruent to any other triangle. Thus, the rst two triangles are in the same equivalence class, while the third and fourthtriangles are in their own equivalence class.

elements that are related to one another, forming what are called equivalence classes. Notationally, given a set Xand an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X whichare equivalent to a. It follows from the denition of the equivalence relations that the equivalence classes form apartition of X. The set of equivalence classes is sometimes called the quotient set or the quotient space of X by ~and is denoted by X / ~.When X has some structure, and the equivalence relation is dened with some connection to this structure, thequotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spacesin topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

1.1 Notation and formal denitionAn equivalence relation is a binary relation ~ satisfying three properties:[1]

For every element a in X, a ~ a (reexivity),

For every two elements a and b in X, if a ~ b, then b ~ a (symmetry)

For every three elements a, b, and c in X, if a ~ b and b ~ c, then a ~ c (transitivity).

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2 CHAPTER 1. EQUIVALENCE CLASS

The equivalence class of an element a is denoted [a] and is dened as the set

[a] = fx 2 X j a xgof elements that are related to a by ~. An alternative notation [a]R can be used to denote the equivalence class of theelement a, specically with respect to the equivalence relation R. This is said to be the R-equivalence class of a.The set of all equivalence classes in Xwith respect to an equivalence relationR is denoted asX/R and called XmoduloR (or the quotient set of X by R).[2] The surjective map x 7! [x] from X onto X/R, which maps each element to itsequivalence class, is called the canonical surjection or the canonical projection map.When an element is chosen (often implicitly) in each equivalence class, this denes an injective map called a section. Ifthis section is denoted by s, one has [s(c)] = c for every equivalence class c. The element s(c) is called a representativeof c. Any element of a class may be chosen as a representative of the class, by choosing the section appropriately.Sometimes, there is a section that is more natural than the other ones. In this case, the representatives are calledcanonical representatives. For example, in modular arithmetic, consider the equivalence relation on the integersdened by a ~ b if a b is a multiple of a given integer n, called the modulus. Each class contains a unique non-negative integer smaller than n, and these integers are the canonical representatives. The class and its representativeare more or less identied, as is witnessed by the fact that the notation a mod n may denote either the class or itscanonical representative (which is the remainder of the division of a by n).

1.2 Examples If X is the set of all cars, and ~ is the equivalence relation has the same color as. then one particular equivalenceclass consists of all green cars. X/~ could be naturally identied with the set of all car colors (cardinality ofX/~ would be the number of all car colors).

Let X be the set of all rectangles in a plane, and ~ the equivalence relation has the same area as. For eachpositive real number A there will be an equivalence class of all the rectangles that have area A.[3]

Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their dierence x yis an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all evennumbers, and the other consisting of all odd numbers. Under this relation [7], [9], and [1] all represent thesame element of Z/~.[4]

Let X be the set of ordered pairs of integers (a,b) with b not zero, and dene an equivalence relation ~ on Xaccording to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b) can beidentied with the rational number a/b, and this equivalence relation and its equivalence classes can be used togive a formal denition of the set of rational numbers.[5] The same construction can be generalized to the eldof fractions of any integral domain.

If X consists of all the lines in, say the Euclidean plane, and L ~M means that L andM are parallel lines, thenthe set of lines that are parallel to each other form an equivalence class as long as a line is considered parallelto itself. In this situation, each equivalence class determines a point at innity.

1.3 PropertiesEvery element x of X is a member of the equivalence class [x]. Every two equivalence classes [x] and [y] are eitherequal or disjoint. Therefore, the set of all equivalence classes of X forms a partition of X: every element of X belongsto one and only one equivalence class.[6] Conversely every partition of X comes from an equivalence relation in thisway, according to which x ~ y if and only if x and y belong to the same set of the partition.[7]

It follows from the properties of an equivalence relation that

x ~ y if and only if [x] = [y].

In other words, if ~ is an equivalence relation on a set X, and x and y are two elements of X, then these statementsare equivalent:

1.4. GRAPHICAL REPRESENTATION 3

x y [x] = [y] [x] \ [y] 6= ;:

1.4 Graphical representationAny binary relation can be represented by a directed graph and symmetric ones, such as equivalence relations, byundirected graphs. If ~ is an equivalence relation on a set X, let the vertices of the graph be the elements of X andjoin vertices s and t if and only if s ~ t. The equivalence classes are represented in this graph by the maximal cliquesforming the connected components of the graph.[8]

1.5 InvariantsIf ~ is an equivalence relation on X, and P(x) is a property of elements of X such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be an invariant of ~, or well-dened under the relation ~.A frequent particular case occurs when f is a function from X to another set Y; if f(x1) = f(x2) whenever x1 ~ x2,then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of nite groups. Some authors use compatible with ~" or just respects ~" instead of invariantunder ~".Any function f : X Y itself denes an equivalence relation on X according to which x1 ~ x2 if and only if f(x1)= f(x2). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class [x] is theinverse image of f(x). This equivalence relation is known as the kernel of f.More generally, a function may map equivalent arguments (under an equivalence relation ~X on X) to equivalentvalues (under an equivalence relation ~Y on Y). Such a function is known as a morphism from ~X to ~Y .

1.6 Quotient space in topologyIn topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relationon a topological space using the original spaces topology to create the topology on the set of equivalence classes.In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebraon the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vectorspace formed by taking a quotient group where the quotient homomorphism is a linear map. By extension, in abstractalgebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotientalgebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a groupaction.The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when theorbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroupon the group by left translations, or respectively the left cosets as orbits under right translation.A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the sensesof topology, abstract algebra, and group actions simultaneously.Although the term can be used for any equivalence relations set of equivalence classes, possibly with further structure,the intent of using the term is generally to compare that type of equivalence relation on a setX either to an equivalencerelation that induces some structure on the set of equivalence classes from a structure of the same kind on X, or to theorbits of a group action. Both the sense of a structure preserved by an equivalence relation and the study of invariantsunder group actions lead to the denition of invariants of equivalence relations given above.

1.7 See also Equivalence partitioning, a method for devising test sets in software testing based on dividing the possible

4 CHAPTER 1. EQUIVALENCE CLASS

program inputs into equivalence classes according to the behavior of the program on those inputs Homogeneous space, the quotient space of Lie groups. Transversal (combinatorics)

1.8 Notes[1] Devlin 2004, p. 122

[2] Wolf 1998, p. 178

[3] Avelsgaard 1989, p. 127

[4] Devlin 2004, p. 123

[5] Maddox 2002, pp. 7778

[6] Maddox 2002, p.74, Thm. 2.5.15

[7] Avelsgaard 1989, p.132, Thm. 3.16

[8] Devlin 2004, p. 123

1.9 References Avelsgaard, Carol (1989), Foundations for Advanced Mathematics, Scott Foresman, ISBN 0-673-38152-8 Devlin, Keith (2004), Sets, Functions, and Logic: An Introduction to Abstract Mathematics (3rd ed.), Chapman& Hall/ CRC Press, ISBN 978-1-58488-449-1

Maddox, Randall B. (2002), Mathematical Thinking and Writing, Harcourt/ Academic Press, ISBN 0-12-464976-9

Morash, Ronald P. (1987), Bridge to Abstract Mathematics, Random House, ISBN 0-394-35429-X Wolf, Robert S. (1998), Proof, Logic and Conjecture: A Mathematicians Toolbox, Freeman, ISBN 978-0-7167-3050-7

1.10 Further readingThis material is basic and can be found in any text dealing with the fundamentals of proof technique, such as any ofthe following:

Sundstrom (2003), Mathematical Reasoning: Writing and Proof, Prentice-Hall Smith; Eggen; St.Andre (2006), A Transition to Advanced Mathematics (6th Ed.), Thomson (Brooks/Cole) Schumacher, Carol (1996), Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley,ISBN 0-201-82653-4

O'Leary (2003), The Structure of Proof: With Logic and Set Theory, Prentice-Hall Lay (2001), Analysis with an introduction to proof, Prentice Hall Gilbert; Vanstone (2005), An Introduction to Mathematical Thinking, Pearson Prentice-Hall Fletcher; Patty, Foundations of Higher Mathematics, PWS-Kent Iglewicz; Stoyle, An Introduction to Mathematical Reasoning, MacMillan D'Angelo; West (2000), Mathematical Thinking: Problem Solving and Proofs, Prentice Hall

1.10. FURTHER READING 5

Cupillari, The Nuts and Bolts of Proofs, Wadsworth Bond, Introduction to Abstract Mathematics, Brooks/Cole Barnier; Feldman (2000), Introduction to Advanced Mathematics, Prentice Hall Ash, A Primer of Abstract Mathematics, MAA

Chapter 2

Equivalence relation

This article is about the mathematical concept. For the patent doctrine, see Doctrine of equivalents.In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are

members of the same cell within a set that has been partitioned into cells such that every element of the set is amember of one and only one cell of the partition. The intersection of any two dierent cells is empty; the union ofall the cells equals the original set. These cells are formally called equivalence classes.

2.1 NotationAlthough various notations are used throughout the literature to denote that two elements a and b of a set are equivalentwith respect to an equivalence relation R, the most common are "a ~ b" and "a b", which are used when R is theobvious relation being referenced, and variations of "a ~R b", "a R b", or "aRb" otherwise.

2.2 DenitionA given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reexive, symmetric andtransitive. Equivalently, for all a, b and c in X:

a ~ a. (Reexivity)

if a ~ b then b ~ a. (Symmetry)

if a ~ b and b ~ c then a ~ c. (Transitivity)

X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is dened as[a] = fb 2 X j a bg .

2.3 Examples

2.3.1 Simple example

Let the set fa; b; cg have the equivalence relation f(a; a); (b; b); (c; c); (b; c); (c; b)g . The following sets are equivalenceclasses of this relation:[a] = fag; [b] = [c] = fb; cg .The set of all equivalence classes for this relation is ffag; fb; cgg .

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2.4. CONNECTIONS TO OTHER RELATIONS 7

2.3.2 Equivalence relations

The following are all equivalence relations:

Has the same birthday as on the set of all people. Is similar to on the set of all triangles. Is congruent to on the set of all triangles. Is congruent to, modulo n" on the integers. Has the same image under a function" on the elements of the domain of the function. Has the same absolute value on the set of real numbers Has the same cosine on the set of all angles.

2.3.3 Relations that are not equivalences The relation "" between real numbers is reexive and transitive, but not symmetric. For example, 7 5 doesnot imply that 5 7. It is, however, a partial order.

The relation has a common factor greater than 1 with between natural numbers greater than 1, is reexiveand symmetric, but not transitive. (Example: The natural numbers 2 and 6 have a common factor greater than1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1).

The empty relation R on a non-empty set X (i.e. aRb is never true) is vacuously symmetric and transitive, butnot reexive. (If X is also empty then R is reexive.)

The relation is approximately equal to between real numbers, even if more precisely dened, is not an equiv-alence relation, because although reexive and symmetric, it is not transitive, since multiple small changes canaccumulate to become a big change. However, if the approximation is dened asymptotically, for example bysaying that two functions f and g are approximately equal near some point if the limit of f g is 0 at that point,then this denes an equivalence relation.

2.4 Connections to other relations A partial order is a relation that is reexive, antisymmetric, and transitive. Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set thatis reexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted forone another, a facility that is not available for equivalence related variables. The equivalence classes of anequivalence relation can substitute for one another, but not individuals within a class.

A strict partial order is irreexive, transitive, and asymmetric. A partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reexive if and onlyif for all a X, there exists a b X such that a ~ b.

A reexive and symmetric relation is a dependency relation, if nite, and a tolerance relation if innite. A preorder is reexive and transitive. A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraicstructure, and which respects the additional structure. In general, congruence relations play the role of kernelsof homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many importantcases congruence relations have an alternative representation as substructures of the structure on which theyare dened. E.g. the congruence relations on groups correspond to the normal subgroups.

8 CHAPTER 2. EQUIVALENCE RELATION

2.5 Well-denedness under an equivalence relationIf ~ is an equivalence relation on X, and P(x) is a property of elements of X, such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be well-dened or a class invariant under the relation ~.A frequent particular case occurs when f is a function from X to another set Y ; if x1 ~ x2 implies f(x1) = f(x2) thenf is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of nite groups. The latter case with the function f can be expressed by a commutative triangle. Seealso invariant. Some authors use compatible with ~" or just respects ~" instead of invariant under ~".More generally, a function may map equivalent arguments (under an equivalence relation ~A) to equivalent values(under an equivalence relation ~B). Such a function is known as a morphism from ~A to ~B.

2.6 Equivalence class, quotient set, partitionLet a; b 2 X . Some denitions:

2.6.1 Equivalence classMain article: Equivalence class

A subset Y of X such that a ~ b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalenceclass of X by ~. Let [a] := fx 2 X j a xg denote the equivalence class to which a belongs. All elements of Xequivalent to each other are also elements of the same equivalence class.

2.6.2 Quotient setMain article: Quotient set

The set of all possible equivalence classes of X by ~, denoted X/ := f[x] j x 2 Xg , is the quotient set of X by~. If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient spacefor the details.

2.6.3 ProjectionMain article: Projection (relational algebra)

The projection of ~ is the function : X ! X/ dened by (x) = [x] which maps elements of X into theirrespective equivalence classes by ~.

Theorem on projections:[1] Let the function f: X B be such that a ~ b f(a) = f(b). Then there is aunique function g : X/~ B, such that f = g. If f is a surjection and a ~ b f(a) = f(b), then g is abijection.

2.6.4 Equivalence kernelThe equivalence kernel of a function f is the equivalence relation ~ dened by x y () f(x) = f(y) . Theequivalence kernel of an injection is the identity relation.

2.6.5 PartitionMain article: Partition of a set

2.7. FUNDAMENTAL THEOREM OF EQUIVALENCE RELATIONS 9

A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single elementof P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their unionis X.

Counting possible partitions

Let X be a nite set with n elements. Since every equivalence relation over X corresponds to a partition of X, andvice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which isthe nth Bell number Bn:

Bn =1

e

1Xk=0

kn

k!;

where the above is one of the ways to write the nth Bell number.

2.7 Fundamental theorem of equivalence relationsA key result links equivalence relations and partitions:[2][3][4]

An equivalence relation ~ on a set X partitions X.

Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongsto a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection from the setof all possible equivalence relations on X and the set of all partitions of X.

2.8 Comparing equivalence relationsIf ~ and are two equivalence relations on the same set S, and a~b implies ab for all a,b S, then is said to be acoarser relation than ~, and ~ is a ner relation than . Equivalently,

~ is ner than if every equivalence class of ~ is a subset of an equivalence class of , and thus every equivalenceclass of is a union of equivalence classes of ~.

~ is ner than if the partition created by ~ is a renement of the partition created by .

The equality equivalence relation is the nest equivalence relation on any set, while the trivial relation that makes allpairs of elements related is the coarsest.The relation "~ is ner than " on the collection of all equivalence relations on a xed set is itself a partial orderrelation.

2.9 Generating equivalence relations Given any set X, there is an equivalence relation over the set [XX] of all possible functions XX. Two suchfunctions are deemed equivalent when their respective sets of xpoints have the same cardinality, correspondingto cycles of length one in a permutation. Functions equivalent in this manner form an equivalence class on[XX], and these equivalence classes partition [XX].


An equivalence relation ~ on X is the equivalence kernel of its surjective projection : X X/~.[5] Conversely,any surjection between sets determines a partition on its domain, the set of preimages of singletons in thecodomain. Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are threeequivalent ways of specifying the same thing.

The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of X X)is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given anybinary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containingR. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x1, x2, ..., xn in Xsuch that a = x1, b = xn, and (xi,xi )R or (xi,xi)R, i = 1, ..., n1.

Note that the equivalence relation generated in this manner can be trivial. For instance, the equivalencerelation ~ generated by:

Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y; Any subset of the identity relation on X has equivalence classes that are the singletons of X.

Equivalence relations can construct new spaces by gluing things together. Let X be the unit Cartesian square[0,1] [0,1], and let ~ be the equivalence relation on X dened by a, b [0,1] ((a, 0) ~ (a, 1) (0, b) ~ (1, b)).Then the quotient space X/~ can be naturally identied (homeomorphism) with a torus: take a square piece ofpaper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder soas to glue together its two open ends, resulting in a torus.

2.10 Algebraic structureMuch of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures themathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics asorder relations, the algebraic structure of equivalences is not as well known as that of orders. The former structuredraws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

2.10.1 Group theoryJust as order relations are grounded in ordered sets, sets closed under pairwise supremum and inmum, equivalencerelations are grounded in partitioned sets, which are sets closed under bijections and preserve partition structure.Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hencepermutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathe-matical structure of equivalence relations.Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denotethe set of bijective functions over A that preserve the partition structure of A: x A g G (g(x) [x]). Then thefollowing three connected theorems hold:[6]

~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations,mentionedabove);

Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the parti-tion;

Given a transformation groupG overA, there exists an equivalence relation ~ over A, whose equivalence classesare the orbits of G.[7][8]

In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are theequivalence classes of A under ~.This transformation group characterisation of equivalence relations diers fundamentally from the way lattices char-acterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe

2.11. EQUIVALENCE RELATIONS AND MATHEMATICAL LOGIC 11

A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a setof bijections, A A.Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a~ b (ab1 H). The equivalence classes of ~also called the orbits of the action of H on Gare the right cosetsof H in G. Interchanging a and b yields the left cosets.Proof.[9] Let function composition interpret group multiplication, and function inverse interpret group inverse. ThenG is a group under composition, meaning that x A g G ([g(x)] = [x]), because G satises the following fourconditions:

G is closed under composition. The composition of any two elements of G exists, because the domain andcodomain of any element of G is A. Moreover, the composition of bijections is bijective;[10]

Existence of identity function. The identity function, I(x)=x, is an obvious element of G; Existence of inverse function. Every bijective function g has an inverse g1, such that gg1 = I; Composition associates. f(gh) = (fg)h. This holds for all functions over all domains.[11]

Let f and g be any two elements of G. By virtue of the denition of G, [g(f(x))] = [f(x)] and [f(x)] = [x], so that[g(f(x))] = [x]. Hence G is also a transformation group (and an automorphism group) because function compositionpreserves the partitioning of A. Related thinking can be found in Rosen (2008: chpt. 10).

2.10.2 Categories and groupoidsLet G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing thisequivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there existsa unique morphism from x to y if and only if x~y.The advantages of regarding an equivalence relation as a special case of a groupoid include:

Whereas the notion of free equivalence relation does not exist, that of a free groupoid on a directed graphdoes. Thus it is meaningful to speak of a presentation of an equivalence relation, i.e., a presentation of thecorresponding groupoid;

Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notionof groupoid, a point of view that suggests a number of analogies;

In many contexts quotienting, and hence the appropriate equivalence relations often called congruences, areimportant. This leads to the notion of an internal groupoid in a category.[12]

2.10.3 LatticesThe possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called ConX by convention. The canonical map ker: X^X Con X, relates the monoid X^X of all functions on X and Con X.ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f: XX toits kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.

2.11 Equivalence relations and mathematical logicEquivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation withexactly two innite equivalence classes is an easy example of a theory which is -categorical, but not categorical forany larger cardinal number.An implication of model theory is that the properties dening a relation can be proved independent of each other(and hence necessary parts of the denition) if and only if, for each property, examples can be found of relationsnot satisfying the given property while satisfying all the other properties. Hence the three dening properties ofequivalence relations can be proved mutually independent by the following three examples:


Reexive and transitive: The relation on N. Or any preorder; Symmetric and transitive: The relation R on N, dened as aRb ab 0. Or any partial equivalence relation; Reexive and symmetric: The relation R on Z, dened as aRb "a b is divisible by at least one of 2 or 3.Or any dependency relation.

Properties denable in rst-order logic that an equivalence relation may or may not possess include:

The number of equivalence classes is nite or innite; The number of equivalence classes equals the (nite) natural number n; All equivalence classes have innite cardinality; The number of elements in each equivalence class is the natural number n.

2.12 Euclidean relationsEuclid's The Elements includes the following Common Notion 1":

Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing equal by are in relationwith). By relation is meant a binary relation, in which aRb is generally distinct from bRa. An Euclidean relationthus comes in two forms:

(aRc bRc) aRb (Left-Euclidean relation)(cRa cRb) aRb (Right-Euclidean relation)

The following theorem connects Euclidean relations and equivalence relations:

Theorem If a relation is (left or right) Euclidean and reexive, it is also symmetric and transitive.

Proof for a left-Euclidean relation

(aRc bRc) aRb [a/c] = (aRa bRa) aRb [reexive; erase T] = bRa aRb. Hence R is symmetric.

(aRc bRc) aRb [symmetry] = (aRc cRb) aRb. Hence R is transitive.

with an analogous proof for a right-Euclidean relation. Hence an equivalence relation is a relation that is Euclideanand reexive. The Elements mentions neither symmetry nor reexivity, and Euclid probably would have deemed thereexivity of equality too obvious to warrant explicit mention.

2.13 See also Partition of a set Equivalence class Up to Conjugacy class Topological conjugacy

2.14. NOTES 13

2.14 Notes[1] Garrett Birkho and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 35, Th. 19. Chelsea.

[2] Wallace, D. A. R., 1998. Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.

[3] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. p. 3, Prop. 2. John Wiley & Sons.

[4] Karel Hrbacek & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, pages 2932, Marcel Dekker

[5] Garrett Birkho and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 33, Th. 18. Chelsea.

[6] Rosen (2008), pp. 243-45. Less clear is 10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press.

[7] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.

[8] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.

[9] Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press: 246.



[12] Borceux, F. and Janelidze, G., 2001. Galois theories, Cambridge University Press, ISBN 0-521-80309-8

2.15 References Brown, Ronald, 2006. Topology and Groupoids. Booksurge LLC. ISBN 1-4196-2722-8. Castellani, E., 2003, Symmetry and equivalence in Brading, Katherine, and E. Castellani, eds., Symmetriesin Physics: Philosophical Reections. Cambridge Univ. Press: 422-433.

Robert Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusseshow equivalence relations arise in lattice theory.

Higgins, P.J., 1971. Categories and groupoids. Van Nostrand. Downloadable since 2005 as a TAC Reprint. John Randolph Lucas, 1973. A Treatise on Time and Space. London: Methuen. Section 31. Rosen, Joseph (2008) Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag.Mostly chpts. 9,10.

Raymond Wilder (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axiomsdening equivalence, pp 4850, John Wiley & Sons.

2.16 External links Hazewinkel, Michiel, ed. (2001), Equivalence relation, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Bogomolny, A., "Equivalence Relationship" cut-the-knot. Accessed 1 September 2009 Equivalence relation at PlanetMath Binary matrices representing equivalence relations at OEIS.


Logical matrices of the 52 equivalence relations on a 5-element set (Colored elds, including those in light gray, stand for ones; whiteelds for zeros.)

Chapter 3

Partition of a set

For the partition calculus of sets, see innitary combinatorics.In mathematics, a partition of a set is a grouping of the sets elements into non-empty subsets, in such a way that

A set of stamps partitioned into bundles: No stamp is in two bundles, and no bundle is empty.

every element is included in one and only one of the subsets.

15

16 CHAPTER 3. PARTITION OF A SET

3.1 DenitionA partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of thesesubsets[1] (i.e., X is a disjoint union of the subsets).Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold:[2]

1. P does not contain the empty set.2. The union of the sets in P is equal to X. (The sets in P are said to cover X.)3. The intersection of any two distinct sets in P is empty. (We say the elements of P are pairwise disjoint.)

In mathematical notation, these conditions can be represented as

1. ; /2 P2. SA2P A = X3. if A;B 2 P and A 6= B then A \B = ; ,

where ; is the empty set.The sets in P are called the blocks, parts or cells of the partition.[3]

The rank of P is |X| |P|, if X is nite.

3.2 Examples Every singleton set {x} has exactly one partition, namely { {x} }. For any nonempty set X, P = {X} is a partition of X, called the trivial partition. For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U,namely, {A, UA}.

The set { 1, 2, 3 } has these ve partitions: { {1}, {2}, {3} }, sometimes written 1|2|3. { {1, 2}, {3} }, or 12|3. { {1, 3}, {2} }, or 13|2. { {1}, {2, 3} }, or 1|23. { {1, 2, 3} }, or 123 (in contexts where there will be no confusion with the number).

The following are not partitions of { 1, 2, 3 }: { {}, {1, 3}, {2} } is not a partition (of any set) because one of its elements is the empty set. { {1, 2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one block. { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however, it is a partitionof {1, 2}.

3.3 Partitions and equivalence relationsFor any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from anypartition P of X, we can dene an equivalence relation on X by setting x ~ y precisely when x and y are in the samepart in P. Thus the notions of equivalence relation and partition are essentially equivalent.[4]

The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly oneelement from each part of the partition. This implies that given an equivalence relation on a set one can select acanonical representative element from every equivalence class.

3.4. REFINEMENT OF PARTITIONS 17

3.4 Renement of partitionsA partition of a set X is a renement of a partition of Xand we say that is ner than and that is coarserthan if every element of is a subset of some element of . Informally, this means that is a further fragmentationof . In that case, it is written that .This ner-than relation on the set of partitions of X is a partial order (so the notation "" is appropriate). Each setof elements has a least upper bound and a greatest lower bound, so that it forms a lattice, and more specically (forpartitions of a nite set) it is a geometric lattice.[5] The partition lattice of a 4-element set has 15 elements and isdepicted in the Hasse diagram on the left.Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a nite set cor-responds to a matroid in which the base set of the matroid consists of the atoms of the lattice, the partitions withn 2 singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of acomplete graph. The matroid closure of a set of atomic partitions is the nest common coarsening of them all; ingraph-theoretic terms, it is the partition of the vertices of the complete graph into the connected components of thesubgraph formed by the given set of edges. In this way, the lattice of partitions corresponds to the graphic matroidof the complete graph.Another example illustrates the rening of partitions from the perspective of equivalence relations. If D is the set ofcards in a standard 52-card deck, the same-color-as relation on D which can be denoted ~C has two equivalenceclasses: the sets {red cards} and {black cards}. The 2-part partition corresponding to ~C has a renement that yieldsthe same-suit-as relation ~S, which has the four equivalence classes {spades}, {diamonds}, {hearts}, and {clubs}.

3.5 Noncrossing partitionsA partition of the set N = {1, 2, ..., n} with corresponding equivalence relation ~ is noncrossing provided that forany two 'cells C1 and C2, either all the elements in C1 are < than all the elements in C2 or they are all > than all theelements in C2. In other words: given distinct numbers a, b, c in N, with a < b < c, if a ~ c (they both are in a cellcalled C), it follows that also a ~ b and b ~ c, that is b is also in C. The lattice of noncrossing partitions of a nite sethas recently taken on importance because of its role in free probability theory. These form a subset of the lattice ofall partitions, but not a sublattice, since the join operations of the two lattices do not agree.

3.6 Counting partitionsThe total number of partitions of an n-element set is the Bell number Bn. The rst several Bell numbers are B0 =1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52, and B6 = 203 (sequence A000110 in OEIS). Bell numbers satisfy therecursion Bn+1 =

Pnk=0

nk

Bk

and have the exponential generating function

1Xn=0

Bnn!

zn = eez1:

The Bell numbers may also be computed using the Bell triangle in which the rst value in each row is copied fromthe end of the previous row, and subsequent values are computed by adding the two numbers to the left and above leftof each position. The Bell numbers are repeated along both sides of this triangle. The numbers within the trianglecount partitions in which a given element is the largest singleton.The number of partitions of an n-element set into exactly k nonempty parts is the Stirling number of the second kindS(n, k).The number of noncrossing partitions of an n-element set is the Catalan number Cn, given by

Cn =1

n+ 1

2n

n

:


3.7 See also Exact cover Cluster analysis Weak ordering (ordered set partition) Equivalence relation Partial equivalence relation Partition renement List of partition topics Lamination (topology)

Rhyme schemes by set partition

3.8 Notes[1] Naive Set Theory (1960). Halmos, Paul R. Springer. p. 28. ISBN 9780387900926.

[2] Lucas, John F. (1990). Introduction to Abstract Mathematics. Rowman & Littleeld. p. 187. ISBN 9780912675732.

[3] Brualdi, pp. 4445

[4] Schechter, p. 54

[5] Birkho, Garrett (1995), Lattice Theory, Colloquium Publications 25 (3rd ed.), American Mathematical Society, p. 95,ISBN 9780821810255.

3.9 References Brualdi, Richard A. (2004). Introductory Combinatorics (4th edition ed.). Pearson Prentice Hall. ISBN 0-13-100119-1.

Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8.

3.9. REFERENCES 19

The 52 partitions of a set with 5 elements


1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53 54

The traditional Japanese symbols for the chapters of the Tale of Genji are based on the 52 ways of partitioning ve elements.

3.9. REFERENCES 21

Partitions of a 4-set ordered by renement


Construction of the Bell triangle

Chapter 4

Quotient space (linear algebra)

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by collapsing N tozero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).

4.1 DenitionFormally, the construction is as follows (Halmos 1974, 21-22). Let V be a vector space over a eld K, and let Nbe a subspace of V. We dene an equivalence relation ~ on V by stating that x ~ y if x y N. That is, x is relatedto y if one can be obtained from the other by adding an element of N. From this denition, one can deduce that anyelement of N is related to the zero vector; more precisely all the vectors in N get mapped into the equivalence classof the zero vector.The equivalence class of x is often denoted

[x] = x + N

since it is given by

[x] = {x + n : n N}.

The quotient space V/N is then dened as V/~, the set of all equivalence classes over V by ~. Scalar multiplicationand addition are dened on the equivalence classes by

[x] = [x] for all K, and [x] + [y] = [x+y].

It is not hard to check that these operations are well-dened (i.e. do not depend on the choice of representative).These operations turn the quotient space V/N into a vector space over K with N being the zero class, [0].The mapping that associates to v V the equivalence class [v] is known as the quotient map.

4.2 ExamplesLet X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Ycan be identied with the space of all lines in X which are parallel to Y. That is to say that, the elements of the setX/Y are lines in X parallel to Y. This gives one way in which to visualize quotient spaces geometrically.Another example is the quotient of Rn by the subspace spanned by the rst m standard basis vectors. The space Rnconsists of all n-tuples of real numbers (x1,,xn). The subspace, identied with Rm, consists of all n-tuples such thatthe last n-m entries are zero: (x1,,xm,0,0,,0). Two vectors of Rn are in the same congruence class modulo the

23

24 CHAPTER 4. QUOTIENT SPACE (LINEAR ALGEBRA)

subspace if and only if they are identical in the last nm coordinates. The quotient space Rn/ Rm is isomorphic toRnm in an obvious manner.More generally, if V is an (internal) direct sum of subspaces U andW,

V = U W

then the quotient space V/U is naturally isomorphic toW (Halmos 1974, Theorem 22.1).An important example of a functional quotient space is a Lp space.

4.3 PropertiesThere is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. Thekernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exactsequence

0! U ! V ! V /U ! 0:

If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may beconstructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, thedimension of V is the sum of the dimensions ofU and V/U. If V is nite-dimensional, it follows that the codimensionof U in V is the dierence between the dimensions of V and U (Halmos 1974, Theorem 22.2):

codim(U) = dim(V /U) = dim(V ) dim(U):

Let T : V W be a linear operator. The kernel of T, denoted ker(T), is the set of all x V such that Tx = 0. Thekernel is a subspace of V. The rst isomorphism theorem of linear algebra says that the quotient space V/ker(T)is isomorphic to the image of V in W. An immediate corollary, for nite-dimensional spaces, is the rank-nullitytheorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of theimage (the rank of T).The cokernel of a linear operator T : V W is dened to be the quotient spaceW/im(T).

4.4 Quotient of a Banach space by a subspaceIf X is a Banach space andM is a closed subspace of X, then the quotient X/M is again a Banach space. The quotientspace is already endowed with a vector space structure by the construction of the previous section. We dene a normon X/M by

k[x]kX/M = infm2M

kxmkX :

The quotient space X/M is complete with respect to the norm, so it is a Banach space.

4.4.1 Examples

Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm.Denote the subspace of all functions f C[0,1] with f(0) = 0 byM. Then the equivalence class of some function g isdetermined by its value at 0, and the quotient space C[0,1] / M is isomorphic to R.If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.

4.5. SEE ALSO 25

4.4.2 Generalization to locally convex spacesThe quotient of a locally convex space by a closed subspace is again locally convex (Dieudonn 1970, 12.14.8).Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {p | A} where A is an index set. Let M be a closed subspace, and dene seminorms q by on X/M

q([x]) = infx2[x]

p(x):

Then X/M is a locally convex space, and the topology on it is the quotient topology.If, furthermore, X is metrizable, then so is X/M. If X is a Frchet space, then so is X/M (Dieudonn 1970, 12.11.3).

4.5 See also quotient set quotient group quotient module quotient space (topology)

4.6 References Halmos, Paul (1974), Finite dimensional vector spaces, Springer, ISBN 978-0-387-90093-3. Dieudonn, Jean (1970), Treatise on analysis, Volume II, Academic Press.

Chapter 5

Quotient space (topology)

For quotient spaces in linear algebra, see quotient space (linear algebra).In topology and related areas of mathematics, a quotient space (also called an identication space) is, intuitively

Illustration of quotient space, S2, obtained by gluing the boundary (in blue) of the disk D2 together to a single point.

speaking, the result of identifying or gluing together certain points of a given topological space. The points to beidentied are specied by an equivalence relation. This is commonly done in order to construct new spaces fromgiven ones. The quotient topology consists of all sets with an open preimage under the canonical projection mapthat maps each element to its equivalence class.

26

5.1. DEFINITION 27

5.1 DenitionLet (X, X) be a topological space, and let ~ be an equivalence relation on X. The quotient space, Y = X / ~ is denedto be the set of equivalence classes of elements of X:

Y = f[x] : x 2 Xg = ffv 2 X : v xg : x 2 Xg;equipped with the topology where the open sets are dened to be those sets of equivalence classes whose unions areopen sets in X:

Y =

8

28 CHAPTER 5. QUOTIENT SPACE (TOPOLOGY)

5.4 Properties

Quotient maps q : X Y are characterized among surjective maps by the following property: if Z is any topologicalspace and f : Y Z is any function, then f is continuous if and only if f q is continuous.

X

q

Y Z

f q

fCharacteristic property of the quotient topology

The quotient space X/~ together with the quotient map q : X X/~ is characterized by the following universalproperty: if g : X Z is a continuous map such that a ~ b implies g(a) = g(b) for all a and b in X, then there exists aunique continuous map f : X/~ Z such that g = f q. We say that g descends to the quotient.The continuous maps dened on X/~ are therefore precisely those maps which arise from continuous maps denedon X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). Thiscriterion is constantly used when studying quotient spaces.Given a continuous surjection q : X Y it is useful to have criteria by which one can determine if q is a quotientmap. Two sucient criteria are that q be open or closed. Note that these conditions are only sucient, not necessary.It is easy to construct examples of quotient maps that are neither open nor closed.

5.5. COMPATIBILITY WITH OTHER TOPOLOGICAL NOTIONS 29

5.5 Compatibility with other topological notions Separation

In general, quotient spaces are ill-behaved with respect to separation axioms. The separation propertiesof X need not be inherited by X/~, and X/~ may have separation properties not shared by X.

X/~ is a T1 space if and only if every equivalence class of ~ is closed in X. If the quotient map is open, then X/~ is a Hausdor space if and only if ~ is a closed subset of the productspace XX.

Connectedness If a space is connected or path connected, then so are all its quotient spaces. A quotient space of a simply connected or contractible space need not share those properties.

Compactness If a space is compact, then so are all its quotient spaces. A quotient space of a locally compact space need not be locally compact.

Dimension The topological dimension of a quotient space can be more (as well as less) than the dimension of theoriginal space; space-lling curves provide such examples.

5.6 See also

5.6.1 Topology Topological space Subspace (topology) Product space Disjoint union (topology) Final topology Mapping cone

5.6.2 Algebra Quotient group Quotient space (linear algebra) Quotient category Mapping cone (homological algebra)

5.7 References Willard, Stephen (1970). General Topology. Reading, MA: Addison-Wesley. ISBN 0-486-43479-6. Quotient space at PlanetMath.org.

Chapter 6

Set (mathematics)

This article is about what mathematicians call intuitive or naive set theory. For a more detailed account, see Naiveset theory. For a rigorous modern axiomatic treatment of sets, see Set theory.In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example,

A set of polygons in a Venn diagram

the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectivelythey form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics.Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as afoundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics suchas Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

30

6.1. DEFINITION 31

The German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxesof the Innite.

6.1 Denition

Passage with the original set denition of Georg Cantor

A set is a well dened collection of distinct objects. The objects that make up a set (also known as the elements ormembers of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor,the founder of set theory, gave the following denition of a set at the beginning of his Beitrge zur Begrndung dertransniten Mengenlehre:[1]

A set is a gathering together into a whole of denite, distinct objects of our perception [Anschauung]or of our thoughtwhich are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the sameelements.[2]

There is the image popular, that sets are like boxes containing their elements. But there is a huge dierence betweenboxes and sets. While boxes don't change their identity when objects are removed from or added to them, sets changetheir identity when their elements change. So its better to have the image of a set as the content of an imaginary box:

A set of polygons The same set as a box The set as the content of a box

Cantors denition turned out to be inadequate for formal mathematics; instead, the notion of a set is taken as anundened primitive in axiomatic set theory, and its properties are dened by the ZermeloFraenkel axioms. Themost basic properties are that a set has elements, and that two sets are equal (one and the same) if and only if everyelement of each set is an element of the other.

6.2 Describing setsMain article: Set notation

There are two ways of describing, or specifying the members of, a set. One way is by intensional denition, using arule or semantic description:

32 CHAPTER 6. SET (MATHEMATICS)

A is the set whose members are the rst four positive integers.B is the set of colors of the French ag.

The second way is by extension that is, listing each member of the set. An extensional denition is denoted byenclosing the list of members in curly brackets:

C = {4, 2, 1, 3}D = {blue, white, red}.

One often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance,A = C and B = D.There are two important points to note about sets. First, a set can have two or more members which are identical, forexample, {11, 6, 6}. However, we say that two sets which dier only in that one has duplicate members are in factexactly identical (see Axiom of extensionality). Hence, the set {11, 6, 6} is exactly identical to the set {11, 6}. Thesecond important point is that the order in which the elements of a set are listed is irrelevant (unlike for a sequenceor tuple). We can illustrate these two important points with an example:

{6, 11} = {11, 6} = {11, 6, 6, 11} .

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the rstthousand positive integers may be specied extensionally as

{1, 2, 3, ..., 1000},

where the ellipsis ("...) indicates that the list continues in the obvious way. Ellipses may also be used where sets haveinnitely many members. Thus the set of positive even numbers can be written as {2, 4, 6, 8, ... }.The notation with braces may also be used in an intensional specication of a set. In this usage, the braces have themeaning the set of all .... So, E = {playing card suits} is the set whose four members are , , , and . A moregeneral form of this is set-builder notation, through which, for instance, the set F of the twenty smallest integers thatare four less than perfect squares can be denoted

F = {n2 4 : n is an integer; and 0 n 19}.

In this notation, the colon (":") means such that, and the description can be interpreted as "F is the set of all numbersof the form n2 4, such that n is a whole number in the range from 0 to 19 inclusive. Sometimes the vertical bar("|") is used instead of the colon.

6.3 MembershipMain article: Element (mathematics)

If B is a set and x is one of the objects of B, this is denoted x B, and is read as x belongs to B, or x is an elementof B. If y is not a member of B then this is written as y B, and is read as y does not belong to B.For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F = {n2 4 : n is an integer; and 0 n 19} dened above,

4 A and 12 F; but9 F and green B.

6.3. MEMBERSHIP 33

6.3.1 SubsetsMain article: Subset

If every member of set A is also a member of set B, then A is said to be a subset of B, written A B (also pronouncedA is contained in B). Equivalently, we can write B A, read as B is a superset of A, B includes A, or B contains A. Therelationship between sets established by is called inclusion or containment.If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A B (A is a proper subset of B)or B A (B is a proper superset of A).Note that the expressions A B and B A are used dierently by dierent authors; some authors use them to meanthe same as A B (respectively B A), whereas other use them to mean the same as A B (respectively B A).

AB

A is a subset of B

Example:

The set of all men is a proper subset of the set of all people. {1, 3} {1, 2, 3, 4}. {1, 2, 3, 4} {1, 2, 3, 4}.


The empty set is a subset of every set and every set is a subset of itself:

A. A A.

An obvious but useful identity, which can often be used to show that two seemingly dierent sets are equal:

A = B if and only if A B and B A.

A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets.

6.3.2 Power setsMain article: Power set

The power set of a set S is the set of all subsets of S. Note that the power set contains S itself and the empty setbecause these are both subsets of S. For example, the power set of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2,3}, {1}, {2}, {3}, }. The power set of a set S is usually written as P(S).The power set of a nite set with n elements has 2n elements. This relationship is one of the reasons for the terminologypower set. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 23 = 8elements.The power set of an innite (either countable or uncountable) set is always uncountable. Moreover, the power set ofa set is always strictly bigger than the original set in the sense that there is no way to pair every element of S withexactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)Every partition of a set S is a subset of the powerset of S.

6.4 CardinalityMain article: Cardinality

The cardinality | S | of a set S is the number of members of S. For example, if B = {blue, white, red}, | B | = 3.There is a unique set with no members and zero cardinality, which is called the empty set (or the null set) and isdenoted by the symbol (other notations are used; see empty set). For example, the set of all three-sided squares haszero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is importantin mathematics; indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.Some sets have innite cardinality. The set N of natural numbers, for instance, is innite. Some innite cardinalitiesare greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers.However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the sameas the cardinality of any segment of that line, of the entire plane, and indeed of any nite-dimensional Euclideanspace.

6.5 Special setsThere are some sets that hold great mathematical importance and are referred to with such regularity that they haveacquired special names and notational conventions to identify them. One of these is the empty set, denoted {} or .Another is the unit set {x}, which contains exactly one element, namely x.[2] Many of these sets are represented usingblackboard bold or bold typeface. Special sets of numbers include

P or , denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}. N or , denoting the set of all natural numbers: N = {1, 2, 3, . . .} (sometimes dened containing 0).

6.6. BASIC OPERATIONS 35

Z or , denoting the set of all integers (whether positive, negative or zero): Z = {..., 2, 1, 0, 1, 2, ...}. Q or , denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b: a, b Z, b 0}. For example, 1/4 Q and 11/6 Q. All integers are in this set since every integer a can beexpressed as the fraction a/1 (Z Q).

R or , denoting the set of all real numbers. This set includes all rational numbers, together with all irrationalnumbers (that is, numbers that cannot be rewritten as fractions, such as 2, as well as transcendental numberssuch as , e and numbers that cannot be dened).

C or , denoting the set of all complex numbers: C = {a + bi : a, b R}. For example, 1 + 2i C. H or , denoting the set of all quaternions: H = {a + bi + cj + dk : a, b, c, d R}. For example, 1 + i + 2j k H.

Positive and negative sets are denoted by a superscript - or +. For example + represents the set of positive rationalnumbers.Each of the above sets of numbers has an innite number of elements, and each can be considered to be a propersubset of the sets listed below it. The primes are used less frequently than the others outside of number theory andrelated elds.

6.6 Basic operationsThere are several fundamental operations for constructing new sets from given sets.

6.6.1 Unions

The union of A and B, denoted A B


Main article: Union (set theory)

Two sets can be added together. The union of A and B, denoted by A B, is the set of all things that are membersof either A or B.Examples:

{1, 2} {1, 2} = {1, 2}. {1, 2} {2, 3} = {1, 2, 3}. {1, 2, 3} {3, 4, 5} = {1, 2, 3, 4, 5}

Some basic properties of unions:

A B = B A. A (B C) = (A B) C. A (A B). A A = A. A = A. A B if and only if A B = B.

6.6.2 IntersectionsMain article: Intersection (set theory)

A new set can also be constructed by determining which members two sets have in common. The intersection of Aand B, denoted by A B, is the set of all things that are members of both A and B. If A B = , then A and B aresaid to be disjoint.Examples:

{1, 2} {1, 2} = {1, 2}. {1, 2} {2, 3} = {2}.

Some basic properties of intersections:

A B = B A. A (B C) = (A B) C. A B A. A A = A. A = . A B if and only if A B = A.

6.6.3 ComplementsMain article: Complement (set theory)

Two sets can also be subtracted. The relative complement of B in A (also called the set-theoretic dierence of A andB), denoted by A \ B (or A B), is the set of all elements that are members of A but not members of B. Note that itis valid to subtract members of a set that are not in the set, such as removing the element green from the set {1, 2,3}; doing so has no eect.In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \A is called the absolute complement or simply complement of A, and is denoted by A.Examples:

6.6. BASIC OPERATIONS 37

The intersection of A and B, denoted A B.

{1, 2} \ {1, 2} = . {1, 2, 3, 4} \ {1, 3} = {2, 4}. If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E= E = O.

Some basic properties of complements:

A \ B B \ A for A B. A A = U. A A = . (A) = A. A \ A = . U = and = U. A \ B = A B.

An extension of the complement is the symmetric dierence, dened for sets A, B as

AB = (A nB) [ (B nA):

For example, the symmetric dierence of {7,8,9,10} and {9,10,11,12} is the set {7,8,11,12}.


The relative complementof B in A

6.6.4 Cartesian productMain article: Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. The Cartesianproduct of two sets A and B, denoted by A B is the set of all ordered pairs (a, b) such that a is a member of A andb is a member of B.Examples:

{1, 2} {red, white} = {(1, red), (1, white), (2, red), (2, white)}. {1, 2} {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green) }. {1, 2} {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.

Some basic properties of cartesian products:

A = . A (B C) = (A B) (A C). (A B) C = (A C) (B C).

Let A and B be nite sets. Then

| A B | = | B A | = | A | | B |.

For example,

{a,b,c}{d,e,f}={(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f)}.

6.7. APPLICATIONS 39

The complement of A in U

6.7 ApplicationsSet theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structuresin abstract algebra, such as groups, elds and rings, are sets closed under one or more operations.One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomainB is a subset of the Cartesian product A B. Given this concept, we are quick to see that the set F of all ordered pairs(x, x2), where x is real, is quite familiar. It has a domain set R and a codomain set that is also R, because the set of allsquares is subset of the set of all reals. If placed in functional notation, this relation becomes f(x) = x2. The reasonthese two are equivalent is for any given value, y that the function is dened for, its corresponding ordered pair, (y,y2) is a member of the set F.

6.8 Axiomatic set theoryMain article: Axiomatic set theory

Although initially naive set theory, which denes a set merely as any well-dened collection, was well accepted, itsoon ran into several obstacles. It was found that this denition spawned several paradoxes, most notably:

Russells paradoxIt shows that the set of all sets that do not contain themselves, i.e. the set { x : x is a setand x x } does not exist.

Cantors paradoxIt shows that the set of all sets cannot exist.

The reason is that the phrase well-dened is not very well dened. It was important to free set theory of theseparadoxes because nearly all of mathematics was being redened in terms of set theory. In an attempt to avoid theseparadoxes, set theory was axiomatized based on rst-order logic, and thus axiomatic set theory was born.


The symmetric dierence of A and B

For most purposes however, naive set theory is still useful.

6.9 Principle of inclusion and exclusionMain article: Inclusion-exclusion principle

This principle gives us the cardinality of the union of sets.

jA1 [A2 [A3 [ : : : [Anj =(jA1j+ jA2j+ jA3j+ : : : jAnj)(jA1 \A2j+ jA1 \A3j+ : : : jAn1 \Anj)+: : :+

(1)n1 (jA1 \A2 \A3 \ : : : \Anj)

6.10 De Morgans LawDe Morgan stated two laws about Sets.If A and B are any two Sets then,

(A B) = A B

The complement of A union B equals the complement of A intersected with the complement of B.

(A B) = A B

The complement of A intersected with B is equal to the complement of A union to the complement of B.

6.11. SEE ALSO 41

6.11 See also Set notation Mathematical object Alternative set theory Axiomatic set theory Category of sets Class (set theory) Dense set Family of sets Fuzzy set Internal set Mereology Multiset Naive set theory Principia Mathematica Rough set Russells paradox Sequence (mathematics) Taxonomy Tuple

6.12 Notes[1] EineMenge, ist die Zusammenfassung bestimmter, wohlunterschiedenerObjekte unserer Anschauung oder unseresDenkens

welche Elemente der Menge genannt werden zu einem Ganzen.[2] Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5.

6.13 References Dauben, JosephW., Georg Cantor: His Mathematics and Philosophy of the Innite, Boston: Harvard UniversityPress (1979) ISBN 978-0-691-02447-9.

Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand (1960) ISBN 0-387-90092-6. Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4. Velleman, Daniel, How To Prove It: A Structured Approach, Cambridge University Press (2006) ISBN 978-0-521-67599-4

6.14 External links C2 Wiki Examples of set operations using English operators. Mathematical Sets: Elements, Intersections & Unions, Education Portal Academy


6.15 Text and image sources, contributors, and licenses6.15.1 Text

Equivalence class Source: https://en.wikipedia.org/wiki/Equivalence_class?oldid=667055947Contributors: AxelBoldt, Zundark, Patrick,Michael Hardy, Wshun, Salsa Shark, Revolver, Charles Matthews, Dysprosia, Wolfgang Kufner, Greenrd, Hyacinth, Psychonaut, Naddy,GreatWhiteNortherner, Tobias Bergemann, Giftlite, WiseWoman, Lethe, Fropu, Fuzzy Logic, Noisy, Tibbetts, Liuyao, Rgdboer,Msh210, MattGiuca, Graham87, Salix alba, Mike Segal, Jameshsher, Laurentius, Hede2000, Arthur Rubin, Lunch, SmackBot, Mhss,Nbarth, Javalenok, Lhf, Mets501, Andrew Delong, Egrin, Magioladitis, David Eppstein, VolkovBot, LokiClock, Rjgodoy, Quietbri-tishjim, Dogah, Henry Delforn (old), Sjn28, Classicalecon, Watchduck, Kausikghatak, Addbot, WikiDreamer Bot, Calle, Rinke 80,Erik9bot, HJ Mitchell, WillNess, Igor Yalovecky, Quondum, D.Lazard, Herebo, Wcherowi, Rpglover64, ChrisGualtieri, Brirush, Markviking, A4b3c2d1e0f, Riddleh, Verdana Bold, Addoergosum and Anonymous: 45

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Quotient space (linear algebra) Source: https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)?oldid=655998606 Contribu-tors: Zundark, Michael Hardy, Giftlite, Fropu, Varuna, Oleg Alexandrov, Joriki, Igny, Frigoris, Bluebot, Silly rabbit, Repliedthemock-turtle, Thijs!bot, RobHar, Englebert, VolkovBot, AlleborgoBot, MikeRumex, SieBot, Ideal gas equation, MystBot, Addbot, Ronhjones,Yobot, RibotBOT, Qm2008q, Sawomir Biay, BenzolBot, Quondum, Helpful Pixie Bot and Anonymous: 11

Quotient space (topology) Source: https://en.wikipedia.org/wiki/Quotient_space_(topology)?oldid=660820903Contributors: AxelBoldt,The Anome, Dominus, Charles Matthews, Dysprosia, Jitse Niesen, Tobias Bergemann, Tosha, Giftlite, Fropu, Bobblewik, Bender235,Oleg Alexandrov, Linas, Marudubshinki, Salix alba, Eubot, YurikBot, Zwobot, Lunch, Schizobullet, Bluebot, Nbarth, Mets501, CBM,Tdvance, Thijs!bot, Reminiscenza, Thomasda, Camrn86, LokiClock, Rei-bot, Anonymous Dissident, Subh83, Kromsson, Prashantva,Alexbot, DumZiBoT, ManDay, Addbot, Luckas-bot, Yobot, Xqbot, Dega180, Magmalex, RjwilmsiBot, EmausBot, WikitanvirBot, Con-jugado, Chricho, Quondum, D.Lazard, Behrooz Abazari, Nuermann, Doublethink1984, Majesty of Knowledge, CrazyDewgong, Aritropand Anonymous: 35

Set (mathematics) Source: https://en.wikipedia.org/wiki/Set_(mathematics)?oldid=671842839 Contributors: Damian Yerrick, Axel-Boldt, Lee Daniel Crocker, Archibald Fitzchestereld, Uriyan, Bryan Derksen, Zundark, The Anome, -- April, LA2, XJaM, Toby~enwiki,Toby Bartels, Pb~enwiki, Waveguy, LapoLuchini, Youandme, Olivier, Paul Ebermann, Frecklefoot, Patrick, TeunSpaans, Michael Hardy,Wshun, Booyabazooka, Fred Bauder, Nixdorf, Wapcaplet, TakuyaMurata, CesarB, Mkweise, Iulianu, Docu, Den fjttrade ankan~enwiki,, UserGoogol, Rob Hooft, Jonik, Mxn, Etaoin, Vargenau, Pizza Puzzle, Schneelocke, Ideyal, Charles Matthews, Timwi,Dcoetzee, Ike9898, Dysprosia, Jitse Niesen, Greenrd, Prumpf, Furrykef, Hyacinth, Saltine, Stormie, RealLink, JorgeGG, Phil Boswell,Robbot, Astronautics~enwiki, Fredrik, Altenmann, Peak, Romanm, Mfc, Tobias Bergemann, Centrx, Giftlite, Smjg, var ArnfjrBjarmason, Lethe, Dissident, Waltpohl, Ezhiki, Proslaes, Python eggs, Chameleon, Gubbubu, Leonard Vertighel, Utcursch, Gdr, Knu-tux, Antandrus, BozMo, Mustafaa, Rousearts, Joseph Myers, Crawdaddio, Maximaximax, Wiml, Tothebarricades.tk, Zfr, Sam Hoce-var, Urhixidur, Vivacissamamente, Porges, Corti, PhotoBox, Shahab, Brianjd, Dissipate, Vinoir, EugeneZelenko, Discospinster, Mani1,Paul August, Demaag, Rgdboer, Crislax, Aaronbrick, Bobo192, Army1987, C S, Johnteslade, Blotwell, , Jumbuck,Msh210, Gary, Tablizer, Eric Kvaalen, Andrewpmk, Richard Fannin, EvenT, Arag0rn, CloudNine, Dirac1933, Spambit, Oleg Alexan-drov, Kendrick Hang, Scndlbrkrttmc, Ott, OwenX, Prashanthns, Marudubshinki, LimoWreck, Graham87, BD2412, Dpr, Josh Par-ris, Mayumashu, MarSch, Trlovejoy, Salix alba, Heah, Bubba73, VKokielov, Mathbot, Crazycomputers, Jrtayloriv, Fresheneesz, Kri,Chobot, Algebraist, YurikBot, Wavelength, Xcelerate, Charles Gaudette, RussBot, Lucinos~enwiki, Thane, Trovatore, Srinivasasha,BOT-Superzerocool, Bota47, RyanJones, Ms2ger, Hirak 99, Lt-wiki-bot, Arthur Rubin, Gulliveig, Reyk, ArielGold, Finell, Dudzcom,RupertMillard, SmackBot, Unyoyega, Bomac, Brick Thrower, BiT, Gilliam, Kaiserb, @modi, Trebor, MartinPoulter, Jerome CharlesPotts, Octahedron80, DHN-bot~enwiki, Bob K, Cybercobra, Nakon, Jiddisch~enwiki, Richard001, MathStatWoman, Just plain Bill,RayGates, SashatoBot, Dfass, Loadmaster, Squigglet, Rosejn, IvanLanin, Tawkerbot2, JRSpriggs, KNM, CRGreathouse, Benjistern,

6.15. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 43

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