Binary relation

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"Relation (mathematics)" redirects here. For a binary relation over a single set (a special case), see § Homogeneous relation. For a more general notion of relation, see finitary relation. For other uses, see Relation (disambiguation).

In mathematics, a binary relation over two sets A and B is a set of ordered pairs (a, b), consisting of elements a of A and elements b of B. That is, it is a subset of the Cartesian product A × B. It encodes the information of relation: an element a is related to an element b, if and only if the pair (a, b) belongs to the set. Binary relation is the most studied form of relations among all n-ary relations.[1]

An example of a binary relation is the "divides" relation over the set of prime numbers P and the set of integers Z, in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.

Binary relations are used in many branches of mathematics to model a wide variety of concepts. These include, among others:

A function may be defined as a special kind of binary relation.[2] Binary relations are also heavily used in computer science.

A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × ⋯ × An, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain Aj of the relation.[3] An example of a ternary relation over Z is "… lies between … and …", which contains triples such as (5, 2, 8), (5, 8, 2) and (−4, 9, −7).

A binary relation over A and B is an element of the power set of A × B. Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of A × B.

As part of set theory, relations are manipulated with the algebra of sets, including complementation. Furthermore, the two sets are considered symmetrically by introduction of the converse relation, which exchanges their places. Another operation is composition of relations. Altogether these tools form the calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.

In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.

The terms correspondence,[4] dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product A × B without reference to A and B, and reserve the term "correspondence" for a binary relation with reference to A and B.

Definition[edit]

Given two sets X and Y, the Cartesian product X × Y is defined as {(x, y) | x ∈ X and y ∈ Y}, and its elements are called ordered pairs.

A binary relation R on X and Y is a subset of X × Y; that is, it is a set of ordered pairs (x, y) consisting of elements x ∈ X and y ∈ Y.[5][note 1] The set X is called the set of departure, and the set Y the set of destination or codomain. (In order to specify the choices of the sets X and Y, some authors define a binary relation or a correspondence as an ordered triple (X, Y, R), where R is a subset of X × Y.) The statement (x, y) ∈ R reads "x is R-related to y", and is denoted by xRy.

When X = Y, a binary relation is called a homogeneous relation. To emphasize the fact X and Y are allowed to be different, a binary relation is also called a heterogeneous relation.[6][7][8] An example of a homogeneous relation is the relation of kinship, where the relation is over people. Homogeneous relation may be viewed as directed graphs, and in the symmetric case as ordinary graphs. Homogeneous relations also encompass orderings as well as partitions of a set (called equivalence relations).

In a binary relation, the order of the elements is important; if a ≠ b then aRb, but bRa can be true or false independently of aRb. For example, 3 divides 9, but 9 does not divide 3.

The domain of R is the set of all x such that xRy for at least one y. The range or image of R is the set of all y such that xRy for at least one x. The field of R is the union of its domain and its range.[9][10][11]

Some authors also call a binary relation a multivalued function;[citation needed] in fact, a (single-valued) partial function from X to Y is nothing but a binary relation over X and Y such that xRy and xRz ⇒ y = z for all x in X and y, z in Y.

Example[edit]

2nd example relation
ball car doll cup
John + − − −
Mary − − + −
Venus − + − −
1st example relation
ball car doll cup
John + − − −
Mary − − + −
Ian − − − −
Venus − + − −

The following example shows that the choice of codomain is important. Suppose there are four objects A = {ball, car, doll, cup} and four people B = {John, Mary, Ian, Venus}. A possible relation on A and B is the relation "is owned by", given by R = {(ball, John), (doll, Mary), (car, Venus)}. That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing. As a set, R does not involve Ian, and therefore R could have been viewed as a subset of A × {John, Mary, Venus}, i.e. a relation over A and {John, Mary, Venus}.

Special types of binary relations[edit]

Example relations over the real numbers.
  y = x2.
  y = 2x + 20.

Some important types of binary relations R over two sets X and Y are listed below.

Uniqueness properties:

  • Injective (also called left-unique[12]): for all x and z in X and y in Y, if xRy and zRy then x = z. For example, the green relation in the diagram is injective, but the red relation is not, as, e.g., it relates both −5 and 5 to 25.
  • Functional (also called right-unique[12], right-definite[13] or univalent[14]): for all x in X, and y and z in Y, if xRy and xRz then y = z. Such a relation is called a partial function. Both relations in the picture are functional. An example of a non-functional relation can be obtained by rotating the red graph clockwise by 90 degrees, i.e. by considering the relation x = y2 which, e.g., relates 25 to both −5 and 5.
  • One-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties (only definable if the sets of departure X resp. destination Y are specified):

  • Left-total:[12] for all x in X there exists a y in Y such that xRy; such an R is also called a multivalued function by some authors.[citation needed] This property, although also referred to as total by some authors,[citation needed] is different from the definition of total in the section Properties. Both relations in the picture are left-total. The relation x = y2, obtained from the above rotation, is not left-total, as, for example, it doesn't relate −14 to any real number.
  • Surjective (also called right-total[12] or onto): for all y in Y, there exists an x in X such that xRy. The green relation is surjective, but the red relation is not, as, for example, it doesn't relate any real number to −14.

Uniqueness and totality properties:

  • A function: a relation that is functional and left-total. Both the green and the red relation are functions.
  • An injective function or injection: a relation that is injective, functional, and left-total.
  • A surjective function or surjection: a relation that is functional, left-total, and right-total.
  • A bijection: a function that is surjective one-to-one (or surjective injective) is said to be bijective, also known as one-to-one correspondence.[15] The green relation is bijective, but the red is not.

Operations on binary relations[edit]

If R and S are binary relations over two sets X and Y, then each of the following is a binary relation over X and Y:

  • Union: R ∪ S ⊆ X × Y, defined as R ∪ S = {(x, y) | (x, y) ∈ R or (x, y) ∈ S}. The identity element is the empty relation. For example, ≥ is the union of > and =.
  • Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = {(x, y) | (x, y) ∈ R and (x, y) ∈ S}. The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisble by 3" and "is divisble by 2".

If R is a binary relation over X and Y, and S is a binary relation over Y and Z then the following is a binary relation over X and Z (for more, see the main article composition of relations):

  • Composition: S ∘ R, also denoted R ; S (or R ∘ S), defined as S ∘ R = {(x, z) | ∃y ∈ Y, (x, y) ∈ R and (y, z) ∈ S}. The identity element is the identity relation. The order of R and S in the notation S ∘ R, used here agrees with the standard notational order for composition of functions. For example, the composition "is mother of" ∘ "is parent of" yields "is maternal grandparent of", while the composition "is parent of" ∘ "is mother of" yields "is grandmother of".

A relation R over two sets X and Y is said to be contained in a relation S over X and Y, written R ⊆ S, if R is a subset of S, that is, for all x in X and y in Y, if xRy, then xSy. If R is contained in S and S is contained in R, then R and S are called equal written R = S. If R is contained in S but S is not contained in R, then R is said to be smaller than S, written R ⊊ S. For example on the rational numbers, the relation > is smaller than ≥, and equal to the composition (> ∘ >).

If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

  • Converse: RT, defined as RT = {(y, x) | (x, y) ∈ R}. A binary relation R (over a set) is equal to its converse RT, if and only if it is symmetric. See also duality (order theory). For example, "is less than" (<) is the converse of "is greater than" (>).

Complement[edit]

If R is a binary relation in X × Y then it has a complementary relation S defined as xSy ⇔ ¬xRy.

An overline or bar is used to indicate the complementary relation: Alternatively, a strikethrough is used to denote complements, for example, = and ≠ are complementary to each other, as are ∈ and ∉, and ⊇ and ⊉. Some authors even use and .[citation needed] In total orderings < and ≥ are complements, as are > and ≤.

The complement of the converse relation RT is the converse of the complement:

If X = Y, the complement has the following properties:

  • If a relation is symmetric, then so is the complement.
  • The complement of a reflexive relation is irreflexive—and vice versa.
  • The complement of a strict weak order is a total preorder—and vice versa.

Restriction[edit]

The restriction of a binary relation over a set X to a subset S is the set of all pairs (x, y) in the relation for which x and y are in S.

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so are its restrictions too.

However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother of the woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.

Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation ≤ is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ≤ to the rational numbers.

The left-restriction (right-restriction, respectively) of a binary relation over two sets X and Y to a subset S of its domain (codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

Matrix representation[edit]

Binary relations over two sets X and Y can be represented algebraically by matrices indexed by X and Y with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND), matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over X and Y and a relation over Y and Z),[16] the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. If X equals Y then the endorelations form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring), and the identity matrix corresponds to the identity relation.[17]

Sets versus classes[edit]

Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory. For example, if we try to model the general concept of "equality" as a binary relation =, we must take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.

In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set A, that contains all the objects of interest, and work with the restriction =A instead of =. Similarly, the "subset of" relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set of a specific set A): the resulting set relation can be denoted ⊆A. Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown that assuming ∈ to be defined over all sets leads to a contradiction in naive set theory.

Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.)[18] With this definition one can for instance define a binary relation over every set and its power set.

Homogeneous relation[edit]

A homogeneous relation (also called endorelation) over a set X is a binary relation over the set X and itself, i.e. it is a subset of the Cartesian product X × X.[8][19][20] It is also simply called a binary relation over X.

A homogeneous relation R over a set X may be identified with a directed simple graph permitting loops, where X is the vertex set and R is the edge set (there is an edge from a vertex x to a vertex y if and only if xRy). The homogenous relation is called the adjacency relation of the directed graph.

The set of all binary relations over a set X is the set 2X × X which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on it forms an inverse semigroup.

Particular homogeneous relations[edit]

Some important particular binary relations over a set X are:

  • the empty relation E = ∅ ⊆ X × X,
  • the universal relation U = X × X, and
  • the identity relation I = {(x, x) | x ∈ X}.

For arbitrary elements x and y of X,

  • xEy holds never,
  • xUy holds always, and
  • xIy holds if and only if x = y.

Properties[edit]

Implications and conflicts between properties of homogenous binary relations
Implications (blue) and conflicts (red) between properties (yellow) of homogenous binary relations. For example, every asymmetric relation is irreflexive ("ASym ⇒ Irrefl"), and no relation on a non-empty set can be both irreflexive and reflexive ("Irrefl # Refl"). Omitting the red edges results in a Hasse diagram.

Some important properties that a binary relation R over a set X may have are:

  • Reflexive: for all x in X, xRx. For example, ≥ is a reflexive relation but > is not.
  • Irreflexive (or strict): for no x in X, xRx. For example, > is an irreflexive relation, but ≥ is not.
  • Coreflexive: for all x and y in X, if xRy then x = y.[21] For example, the relation over the integers in which each odd number is related to itself is a coreflexive relation. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation.
  • Quasi-reflexive: for all x and y in X, if xRy then xRx and yRy.

The previous 4 alternatives are far from being exhaustive; e.g., the red relation y = x2 from the above picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair (0, 0), and (2, 4), but not (2, 2), respectively. The latter two facts also rule out quasi-reflexivity.

  • Symmetric: for all x and y in X, if xRy then yRx. For example, "is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x.
  • Antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false).[22]
  • Asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[23] For example, > is an asymmetric relation, but ≥ is not.

Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation xRy defined by x > 2 is neither symmetric nor antisymmetric, let alone asymmetric.

  • Transitive: for all x, y and z in X, if xRy and yRz then xRz. A transitive relation is irreflexive if and only if it is asymmetric.[24] For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
  • Connex: for all x and y in X, xRy or yRx (or both). This property is sometimes called "total", which is distinct from the definitions of "total" given in the section Special types of binary relations.
  • Trichotomous: for all x and y in X, exactly one of xRy, yRx or x = y holds. For example, > is a trichotomous relation, while the relation "divides" over the natural numbers is not.[25]
  • Right Euclidean (or just Euclidean): for all x, y and z in X, if xRy and xRz then yRz. For example, = is a Euclidean relation because if x = y and x = z then y = z.
  • Left Euclidean: for all x, y and z in X, if yRx and zRx then yRz.
  • Serial: for all x in X, there exists y in X such that xRy. For example, > is a serial relation over the integers. But it is not a serial relation over the positive integers, because there is no y in the positive integers such that 1 > y.[26] However, < is a serial relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given x, choose y = x.
  • Set-like[citation needed] (or local):[citation needed] for all x in X, the class of all y such that yRx is a set. (This makes sense only if relations over proper classes are allowed.) For example, the usual ordering < over the class of ordinal numbers is a set-like relation, while its inverse > is not.
  • Well-founded: every nonempty subset S of X contains a minimal element with respect to R. Well-foundedness implies the descending chain condition (that is, no infinite chain ... xnR...Rx3Rx2Rx1 can exist). If the axiom of dependent choice is assumed, both conditions are equivalent.[27][28]

A preorder is a relation that is reflexive and transitive. A total preorder, also called weak order, is a relation that is reflexive, transitive, and connex. A partial order is a relation that is reflexive, antisymmetric, and transitive. A total order, also called linear order, simple order, connex order, or chain is a relation that is reflexive, antisymmetric, transitive and connex.[29]

A partial equivalence relation is a relation that is symmetric and transitive. An equivalence relation is a relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity.

Operations on homogeneous relations[edit]

If R is a homogeneous relation over X then each of the following is a homogeneous relation over X:

  • Reflexive closure: R=, defined as R= = {(x, x) | x ∈ X} ∪ R or the smallest reflexive relation over X containing R. This can be proven to be equal to the intersection of all reflexive relations containing R.
  • Reflexive reduction: R≠, defined as R≠ = R \ {(x, x) | x ∈ X} or the largest irreflexive relation over X contained in R.
  • Transitive closure: R+, defined as the smallest transitive relation over X containing R. This can be seen to be equal to the intersection of all transitive relations containing R.
  • Reflexive transitive closure: R*, defined as R* = (R+)=, the smallest preorder containing R.
  • Reflexive transitive symmetric closure: R≡, defined as the smallest equivalence relation over X containing R.

All operations defined in the above section #Operations on binary relations also apply to homogeneous relations.

Binary endorelations by property
Reflexivity Symmetry Transitivity Symbol Example
Directed graph →
Undirected graph Symmetric
Tournament Irreflexive Antisymmetric Pecking order
Dependency Reflexive Symmetric
Preorder Reflexive Yes ≤ Preference
Strict preorder Irreflexive Yes <
Total preorder Reflexive Yes ≤
Partial order Reflexive Antisymmetric Yes ≤ Subset
Strict partial order Irreflexive Antisymmetric Yes < Proper subset
Strict weak order Irreflexive Antisymmetric Yes <
Total order Reflexive Antisymmetric Yes ≤
Partial equivalence relation Symmetric Yes
Equivalence relation Reflexive Symmetric Yes ∼, ≅, ≈, ≡ Equality

The number of homogeneous relations[edit]

The number of distinct binary relations over an n-element set is 2n2 (sequence A002416 in the OEIS):

Number of n-element binary relations of different types
Elem­ents Any Transitive Reflexive Preorder Partial order Total preorder Total order Equivalence relation
0 1 1 1 1 1 1 1 1
1 2 2 1 1 1 1 1 1
2 16 13 4 4 3 3 2 2
3 512 171 64 29 19 13 6 5
4 65,536 3,994 4,096 355 219 75 24 15
n 2n2 2n2−n ∑n
k=0  k! S(n, k) 		n! ∑n
k=0  S(n, k)
OEIS A002416 A006905 A053763 A000798 A001035 A000670 A000142 A000110

Notes:

  • The number of irreflexive relations is the same as that of reflexive relations.
  • The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.
  • The number of strict weak orders is the same as that of total preorders.
  • The total orders are the partial orders that are also total preorders. The number of preorders that are neither a partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders, minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.
  • The number of equivalence relations is the number of partitions, which is the Bell number.

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its own complement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse complement).

Examples of common homogeneous relations[edit]

See also[edit]

Notes[edit]

  1. ^ the set R is also sometimes called the graph of the relation R.

References[edit]

  1. ^ "The Definitive Glossary of Higher Mathematical Jargon — Relation". Math Vault. 2019-08-01. Retrieved 2019-12-11.
  2. ^ "Relation definition - Math Insight". mathinsight.org. Retrieved 2019-12-11.
  3. ^ "Relation - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-11.
  4. ^ Jacobson, Nathan (2009), Basic Algebra II (2nd ed.) § 2.1.
  5. ^ Enderton 1977, Ch 3. pg. 40
  6. ^ Schmidt, Gunther; Ströhlein, Thomas (2012). Relations and Graphs: Discrete Mathematics for Computer Scientists. Definition 4.1.1.: Springer Science & Business Media. ISBN 978-3-642-77968-8.CS1 maint: location (link)
  7. ^ Christodoulos A. Floudas; Panos M. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science & Business Media. pp. 299–300. ISBN 978-0-387-74758-3.
  8. ^ a b Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN 978-1-4020-6164-6.
  9. ^ Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN 0-486-61630-4.
  10. ^ Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.
  11. ^ Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979]. Basic Set Theory. Dover. ISBN 0-486-42079-5.
  12. ^ a b c d Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:
    • Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.
    • Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.
    • Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the High Level Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.
  13. ^ Mäs, Stephan (2007), "Reasoning on Spatial Semantic Integrity Constraints", Spatial Information Theory: 8th International Conference, COSIT 2007, Melbourne, Australia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science, 4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18
  14. ^ Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5
  15. ^ Note that the use of "correspondence" here is narrower than as general synonym for binary relation.
  16. ^ John C. Baez (6 Nov 2001). "quantum mechanics over a commutative rig". Newsgroup: sci.physics.research. Usenet: 9s87n0$iv5@gap.cco.caltech.edu. Retrieved November 25, 2018.
  17. ^ Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1, pp. 7-10
  18. ^ Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p. 3. ISBN 0-8218-1041-3.
  19. ^ M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.
  20. ^ Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. Springer Science & Business Media. p. 496. ISBN 978-3-540-67995-0.
  21. ^ Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions to Hash Tables. In Mathematics of Program Construction (p. 337).
  22. ^ Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), A Transition to Advanced Mathematics (6th ed.), Brooks/Cole, p. 160, ISBN 0-534-39900-2
  23. ^ Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer-Verlag, p. 158.
  24. ^ Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School of Mathematics – Physics Charles University. p. 1. Archived from the original (PDF) on 2013-11-02. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
  25. ^ Since neither 5 divides 3, nor 3 divides 5, nor 3=5.
  26. ^ Yao, Y.Y.; Wong, S.K.M. (1995). "Generalization of rough sets using relationships between attribute values" (PDF). Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..
  27. ^ "Condition for Well-Foundedness". ProofWiki. Retrieved 20 February 2019.
  28. ^ Fraisse, R. (15 December 2000). Theory of Relations, Volume 145 - 1st Edition (1st ed.). Elsevier. p. 46. ISBN 9780444505422. Retrieved 20 February 2019.
  29. ^ Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

Bibliography[edit]

External links[edit]

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