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Look up binary in Wiktionary, the free dictionary.

Binary may refer to:

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one). The base-2 numeral system is a positional notation with a radix of 2.

1Science and technology

Science and technology[edit]

Binary Domain Wiki is a FANDOM Games Community. View Mobile Site Gamer Movie Deadpool 2 Honest Trailers.
Just ended the sp campaign and it was a BLAST. Each level was better than the previous. Story is good indeed, and sometimes i was going to cry for Faye. So, Sega, why don't you do a sequel, as promised at the end of this game? This is REALLY a good game. Enjoyed it a LOT.

Mathematics[edit]

Binary number, a representation of numbers using only two digits (0 and 1)
Binary relation, a relation involving two elements
Binary function, a function that takes two arguments
Binary operation, a mathematical operation that takes two arguments
Finger binary, a system for counting in binary numbers on the fingers of human hands

Binary Domain 2: Outlaws is a sequel to first Binary Domain which came out in 2012, a game that Paul still needs to platinum. The story takes place 4 years Paul got the platinum for Binary Domain 1, but he won't do it any time soon, probably have to wait until 2080 for him to do it. Binary Domain is an original squad-based shooter by Toshihiro Nagoshi, the creator behind the Yakuza series. 20-80° (vertical). The slider text always shows 20 degrees when the configuration tool is re-opened (the actual setting is remembered correctly despite this).

Computing[edit]

Binary code, the digital representation of text and data
Binary file, composed of something other than human-readable text
- Executable, a type of binary file that contains machine code for the computer to execute
Binary image, a digital image that has only two possible values for each pixel
Binary prefix, a prefix attached before a unit symbol to multiply it by a power of 2, e.g. Kbyte
Binary tree, a computer tree data structure in which each node has at most two children

Astronomy[edit]

Binary star, a star system with two stars in it
Binary planet, two planetary bodies of comparable mass orbiting each other
Binary asteroid, two asteroids orbiting each other

Chemistry[edit]

Binary compound, a chemical compound containing two different chemical elements
- Binary explosive, an explosive made of two components that become explosive when mixed
- Binary chemical weapon, containing two chemicals that when combined make a toxic agent

Arts and entertainment[edit]

Binary (Kay Tse album), 2008
Binary (Ani DiFranco album), 2017
Binary (comics), a superheroine in the Marvel Universe
Binary (novel), a 1972 novel by Michael Crichton (writing as John Lange)
'Binary' (song), a 2007 single by Assemblage 23
Binary form, a way of structuring a piece of music
Binary, an evil organization in the novel InterWorld

Other uses[edit]

Binary option, a financial option in which the payoff is either some fixed monetary amount or nothing at all
Binary betting, a bet on a proposition which is quoted as a spread bet
Binary opposition, polar opposites, often ignoring the middle ground
Binary Research, a company founded in Auckland, New Zealand
Gender binary, the classification of sex and gender into two distinct and disconnected forms of masculine and feminine
- Non-binary gender, gender identities outside the gender binary
The Binary, a tower in Dubai, United Arab Emirates

Definition[edit]

Given two sets X and Y, the Cartesian productX × 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.^[2]^{[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 is read '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.^[3]^[4]^[5] An example of a homogeneous relation is a kinship where the relations are 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).

The order of the elements is important; if a ≠ b then aRb and bRa can be true or false independently of each other. 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.^[6]^[7]^[8]

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 '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.

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

Uniqueness properties:

Injective (also called left-unique^[9]): 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^[9], right-definite^[10] or univalent^[11]): for all x in X, and y and z in Y, if xRy and xRz then y = z; such a binary 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:^[9] 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 next section. Both relations in the picture are left-total. The relation x = y², obtained from the above rotation, is not left-total, as, e.g., it doesn't relate −14 to any real number.
Surjective (also called right-total^[9] 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, e.g., 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 surjective one-to-one or surjective injective function is said to be bijective, also known as one-to-one correspondence.^[12] 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.

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: (see 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 if R is a subset of S, that is, for all x in X and y in Y, if xRy then xSy. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is contained in ≥.

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

Converse: R^T, defined as R^T = {(y, x) | (x, y) ∈ R}. A binary relation over a set is equal to^{[clarification needed]} its converse 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 relationS defined as xSy ⇔ ¬xRy.

An overline or bar is used to indicate the complementary relation: ${displaystyle S={bar {R}}.}$ 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 ${displaystyle R}$ and ${displaystyle not R}$ .^{[citation needed]} In total orderings < and ≥ are complements, as are > and ≤.

The complement of the converse relationR^T is the converse of the complement: ${displaystyle {overline {R^{mathsf {T}}}}={bar {R}}^{mathsf {T}}.}$

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

If a relation is symmetric, the complement is too.
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, its restrictions are 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.

Binary Domain All Endings

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),^[13] 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.^[14]

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.)^[15] 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.^[5]^[16]^[17] 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 ${displaystyle {mathcal {B}}(X)}$ over a set X is the set 2^{X × 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 ${displaystyle {mathcal {B}}(X),}$ it forms a inverse semigroup.

Particular homogeneous relations[edit]

Some important particular binary relations over a set X are:

the empty relationE = ∅ ⊆ X × X,
the universal relationU = X × X, and
the identity relationI = {(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.^[18] 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).^[19]
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.^[20] 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.^[21] 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 previous section.
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.^[22]
Right Euclidean (or just Euclidean): for all x, y and z in X, if xRy and xRz then yRz. For example, = is an 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.^[23] 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 … x_nR…Rx₃Rx₂Rx₁ can exist). If the axiom of dependent choice is assumed, both conditions are equivalent.^[24]^[25]

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.^[26]

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 2^n² (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	2^n²	2^n²−n	∑n k=0k! S(n, k)	n!	∑n k=0S(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]

Binary Domain Plot

Order relations, including strict orders:
- Greater than or equal to
- Less than or equal to
- Divides (evenly)
- Subset of
Equivalence relations:
- Parallel with (for affine spaces)
- Is in bijection with
Tolerance relation, a reflexive and symmetric relation:
- Dependency relation, a finite tolerance relation
- Independency relation, the complement of some dependency relation

Notes[edit]

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

^Jacobson, Nathan (2009), Basic Algebra II (2nd ed.) § 2.1.
^Enderton 1977, Ch 3. pg. 40
^Schmidt, Gunther; Ströhlein, Thomas (2012). Relations and Graphs: Discrete Mathematics for Computer Scientists. Definition 4.1.1.: Springer Science & Business Media. ISBN978-3-642-77968-8.
^Christodoulos A. Floudas; Panos M. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science & Business Media. pp. 299–300. ISBN978-0-387-74758-3.
^ ^a^bMichael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.
^Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.
^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. ISBN978-0-486-47484-7.
^Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979]. Basic Set Theory. Dover. ISBN0-486-42079-5.
^ ^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. ISBN978-3-540-67995-0.
- Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN978-0-13-460643-9.
- 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. ISBN978-3-89675-629-9.
^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
^Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN978-0-521-76268-7, Chapt. 5
^Note that the use of 'correspondence' here is narrower than as general synonym for binary relation.
^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.
^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
^Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p. 3. ISBN0-8218-1041-3.
^M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN978-0-521-19021-3.
^Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. Springer Science & Business Media. p. 496. ISBN978-3-540-67995-0.
^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).
^Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), A Transition to Advanced Mathematics (6th ed.), Brooks/Cole, p. 160, ISBN0-534-39900-2
^Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer-Verlag, p. 158.
^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.Cite uses deprecated parameter |deadurl= (help) Lemma 1.1 (iv). This source refers to asymmetric relations as 'strictly antisymmetric'.
^Since neither 5 divides 3, nor 3 divides 5, nor 3=5.
^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..
^'Condition for Well-Foundedness'. ProofWiki. Retrieved 20 February 2019.
^Fraisse, R. (15 December 2000). Theory of Relations, Volume 145 - 1st Edition (1st ed.). Elsevier. p. 46. ISBN9780444505422. Retrieved 20 February 2019.
^Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN0-12-597680-1, p. 4

Binary Domain Review

References[edit]

Binary Domain Walkthrough

Enderton, Herbert (1977), Elements of set theory, Boston, MA: Academic Press, ISBN978-0-12-238440-0.
M. Kilp, U. Knauer, A.V. Mikhalev (2000) Monoids, Acts and Categories: with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, ISBN3-11-015248-7.
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