{"history":[],"article":{"id":"f608a5e8-f50d-4873-ac71-b60ddaa71313","title":"Relation (mathematics)","slug":"relation-mathematics","content":":::summary\n:::\n\n# \n[[File:Relación binaria 01.svg|thumb|300px|Illustration of an example relation on a set\nA =\n. An arrow from x to y indicates that the relation holds between x and y. The relation is represented by the set\n{{math|{ (a,a),}}\n(a,b),\n(a,d),\n(b,a),\n(b,d),\n(c,b),\n(d,c),\n{{math|(d,d) } }} of ordered pairs.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), a **relation** denotes some kind of **relation**ship between two [objects](https://bliptext.com/articles/mathematical-object) in a [set](https://bliptext.com/articles/set-mathematics), which may or may not hold.\nAs an example, \"is less than\" is a relation on the set of [natural numbers](https://bliptext.com/articles/natural-number); it holds, for instance, between the values\n1\nand\n3\n(denoted as\n1 < 3\n), and likewise between\n3\nand\n4\n(denoted as\n3 < 4\n), but not between the values\n3\nand\n1\nnor between\n4\nand\n4\n, that is,\n3 < 1\nand\n4 < 4\nboth evaluate to false.\nAs another example, \"is sister of is a relation on the set of all people, it holds e.g. between [Marie Curie](https://bliptext.com/articles/marie-curie) and [Bronisława Dłuska](https://bliptext.com/articles/bronis-awa-d-uska), and likewise vice versa. Set members may not be in relation \"to a certain degree\" – either they are in relation or they are not.\nFormally, a relation R over a set X can be seen as a set of [ordered pairs](https://bliptext.com/articles/ordered-pair)\n(x,y)\nof members of X. The relation R holds between x and y if\n(x,y)\nis a member of R. For example, the relation \"is less than\" on the natural numbers is an infinite set\nRless\nof pairs of natural numbers that contains both\n(1,3)\nand\n(3,4)\n, but neither\n(3,1)\nnor\n(4,4)\n. The relation \"is a [nontrivial divisor](https://bliptext.com/articles/nontrivial-divisor) of on the set of one-digit natural numbers is sufficiently small to be shown here:\nRdv =\n2\nis a nontrivial divisor of\n8\n, but not vice versa, hence\n(2,8) ∈ Rdv\n, but\n(8,2) ∉ Rdv\n.\nIf R is a relation that holds for x and y one often writes\nxRy\n. For most common relations in mathematics, special symbols are introduced, like \"\n<\n\" for \"is less than\", and \"\n\" for \"is a nontrivial divisor of\", and, most popular \"\n=\n\" for \"is equal to\". For example, \"\n1 < 3\n\", \"\n1\nis less than\n3\n\", and \"\n(1,3) ∈ Rless\n\" mean all the same; some authors also write \"\n(1,3) ∈ (<)\n\".\nVarious properties of relations are investigated.\nA relation R is reflexive if\nxRx\nholds for all x, and irreflexive if\nxRx\nholds for no x. It is symmetric if\nxRy\nalways implies\nyRx\n, and asymmetric if\nxRy\nimplies that\nyRx\nis impossible.\nIt is transitive if\nxRy\nand\nyRz\nalways implies\nxRz\n. For example, \"is less than\" is irreflexive, asymmetric, and transitive, but neither reflexive nor symmetric.\n\"is sister of is transitive, but neither reflexive (e.g. [Pierre Curie](https://bliptext.com/articles/pierre-curie) is not a sister of himself), nor symmetric, nor asymmetric; while being irreflexive or not may be a matter of definition (is every woman a sister of herself?), \"is ancestor of is transitive, while \"is parent of is not. Mathematical theorems are known about combinations of relation properties, such as \"a transitive relation is irreflexive if, and only if, it is asymmetric\".\nOf particular importance are relations that satisfy certain combinations of properties.\nA [partial order](https://bliptext.com/articles/partial-order) is a relation that is reflexive, antisymmetric, and transitive,\nan [equivalence relation](https://bliptext.com/articles/equivalence-relation) is a relation that is reflexive, symmetric, and transitive,\na [function](https://bliptext.com/articles/function-mathematics) is a relation that is right-unique and left-total (see [below](https://bliptext.com/articles/relation-mathematics)).\nSince relations are sets, they can be manipulated using set operations, including [union](https://bliptext.com/articles/union-set-theory), [intersection](https://bliptext.com/articles/intersection-set-theory), and [complementation](https://bliptext.com/articles/complement-set-theory), leading to the [algebra of sets](https://bliptext.com/articles/algebra-of-sets).\nFurthermore, the [calculus of relations](https://bliptext.com/articles/calculus-of-relations) includes the operations of taking the [converse](https://bliptext.com/articles/converse-relation) and [composing relations](https://bliptext.com/articles/composition-of-relations).\nThe above concept of relation has been generalized to admit relations between members of two different sets ([heterogeneous relation](https://bliptext.com/articles/heterogeneous-relation), like \"lies on\" between the set of all [points](https://bliptext.com/articles/point-geometry) and that of all [lines](https://bliptext.com/articles/line-geometry) in geometry), relations between three or more sets ([finitary relation](https://bliptext.com/articles/finitary-relation), like ''\"person\nx\nlives in town\ny\nat time\nz\n\"), and relations between [classes](https://bliptext.com/articles/class-mathematics) (like \"is an element of on the class of all sets, see '').\n\n# Definition\nGiven a set\nX\n, a relation\nR\nover\nX\nis a set of [ordered pairs](https://bliptext.com/articles/ordered-pair) of elements from\nX\n, formally:\nR ⊆\n.\nThe statement\n(x,y) ∈ R\nreads \"\nx\nis\nR\n-related to\ny\n\" and is written in [infix notation](https://bliptext.com/articles/infix-notation) as\nxRy\n. The order of the elements is important; if\nx ≠ y\nthen\nyRx\ncan be true or false independently of\nxRy\n. For example,\n3\ndivides\n9\n, but\n9\ndoes not divide\n3\n.\n\n# Representation of relations\nA relation\nR\non a finite set\nX\nmay be represented as:\nX\ncorresponds to a vertex; a directed edge from\nx\nto\ny\nexists if and only if\n(x,y) ∈ R\n.\nX\nare arranged in some fixed sequence\nx1\n, ...,\nxn\nn × n\n, with the element in line\ni\n, column\nj\n, being Yes check.svg, if\n(xi,xj) ∈ R\n, and Dark Red x.svg, otherwise.\nR\nof [real numbers](https://bliptext.com/articles/real-number) can be represented as a two-dimensional geometric figure: using [Cartesian coordinates](https://bliptext.com/articles/cartesian-coordinates), draw a [point](https://bliptext.com/articles/point-geometry) at\n(x,y)\nwhenever\n(x,y) ∈ R\n. A transitive relation\nR\non a finite set\nX\nmay be also represented as\nX\ncorresponds to a vertex; directed edges are drawn such that a [directed path](https://bliptext.com/articles/directed-path) from\nx\nto\ny\nexists if and only if\n(x,y) ∈ R\n. Compared to a directed-graph representation, a Hasse diagram needs fewer edges, leading to a less tangled image.\nSince the relation \"''a directed path exists from\nx\nto\ny\n''\" is transitive, only transitive relations can be represented in Hasse diagrams. Usually the diagram is laid out such that all edges point in an upward direction, and the arrows are omitted.\nFor example, on the set of all divisors of\n12\n, define the relation\nRdiv\nby\nx Rdiv y\nif\nx\nis a divisor of\ny\nand\nx ≠ y\n. Formally,\nX =\nand\nRdiv =\n. The representation of\nRdiv\nas a Boolean matrix is shown in the middle table; the representation both as a Hasse diagram and as a directed graph is shown in the left picture.\nThe following are equivalent:\nx Rdiv y\nis true.\n(x,y) ∈ Rdiv\n.\nx\nto\ny\nexists in the Hasse diagram representing\nRdiv\n.\nx\nto\ny\nexists in the directed graph representing\nRdiv\n.\nRdiv\n, the element in line\nx\n, column\ny\nis \"Yes check.svg\".\nAs another example, define the relation\nRel\non\nR\nby\nx Rel y\nif\nx2 + xy + y2 = 1\n. The representation of\nRel\nas a 2D-plot obtains an ellipse, see right picture.\nSince\nR\nis not finite, neither a directed graph, nor a finite Boolean matrix, nor a Hasse diagram can be used to depict\nRel\n.\n\n# Properties of relations\nSome important properties that a relation R over a set X may have are:\nx ∈ X\n,\nxRx\n. For example,\n≥\nis a reflexive relation but\n>\nis not.\nx ∈ X\n, not\nxRx\n. For example,\n>\nis an irreflexive relation, but\n≥\nis not.\nThe previous 2 alternatives are not exhaustive; e.g., the red relation\ny = x2\ngiven in the diagram below is neither irreflexive, nor reflexive, since it contains the pair\n(0,0)\n, but not\n(2,2)\n, respectively.\nx, y ∈ X\n, if\nxRy\nthen\nyRx\n. 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.\nx, y ∈ X\n, if\nxRy\nand\nyRx\nthen\nx = y\n. For example,\n≥\nis an antisymmetric relation; so is\n>\n, but [vacuously](https://bliptext.com/articles/vacuous-truth) (the condition in the definition is always false).\nx, y ∈ X\n, if\nxRy\nthen not\nyRx\n. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.\nFor example,\n>\nis an asymmetric relation, but\n≥\nis not.\nAgain, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation\nxRy\ndefined by\nx > 2\nis neither symmetric (e.g.\n5R1\n, but not\n1R5\n) nor antisymmetric (e.g.\n6R4\n, but also\n4R6\n), let alone asymmetric.\nx, y, z ∈ X\n, if\nxRy\nand\nyRz\nthen\nxRz\n. A transitive relation is irreflexive if and only if it is asymmetric.\nFor example, \"is ancestor of\" is a transitive relation, while \"is parent of\" is not.\nx, y ∈ X\n, if\nx ≠ y\nthen\nxRy\nor\nyRx\n. For example, on the natural numbers,\n<\nis connected, while \"is a divisor of is not (e.g. neither\n5R7\nnor\n7R5\n).\nx, y ∈ X\n,\nxRy\nor\nyRx\n. For example, on the natural numbers,\n≤\nis strongly connected, but\n<\nis not.\nA relation is strongly connected if, and only if, it is connected and reflexive.\nUniqueness properties:\nx, y, z ∈ X\n, if\nxRy\nand\nzRy\nthen\nx = z\n. For example, the green and blue relations in the diagram are injective, but the red one is not (as it relates both\n−1\nand\n1\nto\n1\n), nor is the black one (as it relates both\n−1\nand\n1\nto\n0\n).\nx, y, z ∈ X\n, if\nxRy\nand\nxRz\nthen\ny = z\n. Such a relation is called a.\nFor example, the red and green relations in the diagram are functional, but the blue one is not (as it relates\n1\nto both\n−1\nand\n1\n), nor is the black one (as it relates 0 to both −1 and 1).\nTotality properties:\nx ∈ X\n, there exists some\ny ∈ X\nsuch that\nxRy\n. Such a relation is called a [multivalued function](https://bliptext.com/articles/multivalued-function).\nFor example, the red and green relations in the diagram are total, but the blue one is not (as it does not relate\n−1\nto any real number), nor is the black one (as it does not relate\n2\nto any real number). As another example,\n>\nis a serial relation over the integers.\nBut it is not a serial relation over the positive integers, because there is no y in the positive integers such that\n1 > y\n. However,\n<\nis a serial relation over the positive integers, the rational numbers and the real numbers.\nEvery reflexive relation is serial: for a given x, choose\ny = x\n.\ny ∈ Y\n, there exists an\nx ∈ X\nsuch that\nxRy\n. For example, the green and blue relations in the diagram are surjective, but the red one is not (as it does not relate any real number to\n−1\n), nor is the black one (as it does not relate any real number to\n2\n).\n# [Examples of four types of relations over the real numbers: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black). 2D-plot representation is used. | upload.wikimedia.org/wikipedia/commons/b/b6/The///four///types///of///binary///relations.png]\n\n## Combinations of properties\n! ! ! ! ! ! ! [Partial order](https://bliptext.com/articles/partially-ordered-set)\n! [Strict partial order](https://bliptext.com/articles/partially-ordered-set)\n! [Total order](https://bliptext.com/articles/total-order)\n! [Strict total order](https://bliptext.com/articles/total-order)\n! [Equivalence relation](https://bliptext.com/articles/equivalence-relation)\nRelations that satisfy certain combinations of the above properties are particularly useful, and thus have received names by their own.\nOrderings:\nUniqueness properties:\nUniqueness and totality properties:\n\n# Operations on relations\nR\nand\nS\nare relations over\nX\nthen\nR ∪ S =\nis the of\nR\nand\nS\n. The identity element of this operation is the empty relation.\nFor example,\n≤\nis the union of\n<\nand\n=\n, and\n≥\nis the union of\n>\nand\n=\n.\nR\nand\nS\nare relations over\nX\nthen\nR ∩ S =\nis the of\nR\nand\nS\n. The identity element of this operation is the universal relation.\nFor example, \"is a lower card of the same suit as\" is the intersection of \"is a lower card than\" and \"belongs to the same suit as\".\nR\nand\nS\nare relations over\nX\nthen\nS ∘ R =\n(also denoted by\nR; S\n) is the [relative product](https://bliptext.com/articles/relative-product) of\nR\nand\nS\n. The identity element is the identity relation.\nThe order of\nR\nand\nS\nin the notation\nS ∘ R\n, used here agrees with the standard notational order for [composition of functions](https://bliptext.com/articles/composition-of-functions).\nFor example, the composition \"is mother of\"\n∘\n\"is parent of\" yields \"is maternal grandparent of\", while the composition \"is parent of\"\n∘\n\"is mother of\" yields \"is grandmother of\".\nFor the former case, if\nx\nis the parent of\ny\nand\ny\nis the mother of\nz\n, then\nx\nis the maternal grandparent of\nz\n.\nR\nis a relation over sets\nX\nand\nY\nthen\nRT =\nis the converse relation of\nR\nover\nY\nand\nX\n. For example,\n=\nis the converse of itself, as is\n≠\n, and\n<\nand\n>\nare each other's converse, as are\n≤\nand\n≥\n.\nR\nis a relation over\nX\nthen\nR =\n(also denoted by\nR\nor\n¬R\n) is the complementary relation of\nR\n. For example,\n=\nand\n≠\nare each other's complement, as are\n⊆\nand\n⊈\n,\n⊇\nand\n⊉\n, and\n∈\nand\n∉\n, and, for [total orders](https://bliptext.com/articles/total-order), also\n<\nand\n≥\n, and\n>\nand\n≤\n. The complement of the [converse relation](https://bliptext.com/articles/converse-relation)\nRT\nis the converse of the complement:\nR\nis a relation over\nX\nand\nS\nis a subset of\nX\nthen\nRundefined =\nis the of\nR\nto\nS\n. The expression\nRundefined =\nis the of\nR\nto\nS\nRundefined =\nis called the of\nR\nto\nS\n. If a relation is [reflexive](https://bliptext.com/articles/reflexive-relation), ir[reflexive](https://bliptext.com/articles/reflexive-relation), [symmetric](https://bliptext.com/articles/symmetric-relation), anti[symmetric](https://bliptext.com/articles/symmetric-relation), a[symmetric](https://bliptext.com/articles/symmetric-relation), [transitive](https://bliptext.com/articles/transitive-relation), [total](https://bliptext.com/articles/serial-relation), [trichotomous](https://bliptext.com/articles/trichotomy-mathematics), a [partial order](https://bliptext.com/articles/partial-order), [total](https://bliptext.com/articles/serial-relation) order, [strict weak order](https://bliptext.com/articles/strict-weak-order), [total](https://bliptext.com/articles/serial-relation) preorder (weak order), or an [equivalence relation](https://bliptext.com/articles/equivalence-relation), then so too are its restrictions.\nHowever, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal.\nFor example, restricting the relation \"\nx\nis parent of\ny\n\" to females yields the relation \"\nx\nis mother of the woman\ny\n\"; its transitive closure does not 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.\nA relation\nR\nover sets\nX\nand\nY\nis said to be a relation\nS\nover\nX\nand\nY\n, written\nR ⊆ S\n, if\nR\nis a subset of\nS\n, that is, for all\nx ∈ X\nand\ny ∈ Y\n, if\nxRy\n, then\nxSy\n. If\nR\nis contained in\nS\nand\nS\nis contained in\nR\n, then\nR\nand\nS\nare called equal written\nR = S\n. If\nR\nis contained in\nS\nbut\nS\nis not contained in\nR\n, then\nR\nis said to be than\nS\n, written\nR ⊊ S\n. For example, on the [rational numbers](https://bliptext.com/articles/rational-number), the relation\n>\nis smaller than\n≥\n, and equal to the composition\n> ∘ >\n.\n\n# Theorems about relations\nR\nis contained in relation\nS\n, then\nR\nis reflexive, connected, strongly connected, left-total, or right-total, then so is\nS\n.\nS\nis irreflexive, asymmetric, anti-symmetric, left-unique, or right-unique, then so is\nR\n.\n\n# Examples\n\n# Generalizations\nThe above concept of relation has been generalized to admit relations between members of two different sets.\nGiven sets\nX\nand\nY\n, a [heterogeneous relation](https://bliptext.com/articles/heterogeneous-relation)\nR\nover\nX\nand\nY\nis a subset of\n. When\nX = Y\n, the relation concept described above is obtained; it is often called homogeneous relation (or endorelation) to distinguish it from its generalization.\nThe above properties and operations that are marked \"\" and \"\", respectively, generalize to heterogeneous relations.\nAn example of a heterogeneous relation is \"ocean\nx\nborders continent\ny\n\". The best-known examples are [functions](https://bliptext.com/articles/function-mathematics) with distinct domains and ranges, such as\nsqrt : N → R+\n.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T11:40:58.509Z","updated_at":"2025-01-26T11:40:58.509Z","search_vector":"'mathemat':2 'relat':1","search_content":" \n \n\n# \n[[File:Relación binaria 01.svg|thumb|300px|Illustration of an example relation on a set\nA =\n. An arrow from x to y indicates that the relation holds between x and y. The relation is represented by the set\n{{math|{ (a,a),}}\n(a,b),\n(a,d),\n(b,a),\n(b,d),\n(c,b),\n(d,c),\n{{math|(d,d) } }} of ordered pairs.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), a **relation** denotes some kind of **relation**ship between two [objects](https://bliptext.com/articles/mathematical-object) in a [set](https://bliptext.com/articles/set-mathematics), which may or may not hold.\nAs an example, \"is less than\" is a relation on the set of [natural numbers](https://bliptext.com/articles/natural-number); it holds, for instance, between the values\n1\nand\n3\n(denoted as\n1 \nis not.\nx ∈ X\n, not\nxRx\n. For example,\n>\nis an irreflexive relation, but\n≥\nis not.\nThe previous 2 alternatives are not exhaustive; e.g., the red relation\ny = x2\ngiven in the diagram below is neither irreflexive, nor reflexive, since it contains the pair\n(0,0)\n, but not\n(2,2)\n, respectively.\nx, y ∈ X\n, if\nxRy\nthen\nyRx\n. 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.\nx, y ∈ X\n, if\nxRy\nand\nyRx\nthen\nx = y\n. For example,\n≥\nis an antisymmetric relation; so is\n>\n, but [vacuously](https://bliptext.com/articles/vacuous-truth) (the condition in the definition is always false).\nx, y ∈ X\n, if\nxRy\nthen not\nyRx\n. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.\nFor example,\n>\nis an asymmetric relation, but\n≥\nis not.\nAgain, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation\nxRy\ndefined by\nx > 2\nis neither symmetric (e.g.\n5R1\n, but not\n1R5\n) nor antisymmetric (e.g.\n6R4\n, but also\n4R6\n), let alone asymmetric.\nx, y, z ∈ X\n, if\nxRy\nand\nyRz\nthen\nxRz\n. A transitive relation is irreflexive if and only if it is asymmetric.\nFor example, \"is ancestor of\" is a transitive relation, while \"is parent of\" is not.\nx, y ∈ X\n, if\nx ≠ y\nthen\nxRy\nor\nyRx\n. For example, on the natural numbers,\n \nis a serial relation over the integers.\nBut it is not a serial relation over the positive integers, because there is no y in the positive integers such that\n1 > y\n. However,\n \nand\n=\n.\nR\nand\nS\nare relations over\nX\nthen\nR ∩ S =\nis the of\nR\nand\nS\n. The identity element of this operation is the universal relation.\nFor example, \"is a lower card of the same suit as\" is the intersection of \"is a lower card than\" and \"belongs to the same suit as\".\nR\nand\nS\nare relations over\nX\nthen\nS ∘ R =\n(also denoted by\nR; S\n) is the [relative product](https://bliptext.com/articles/relative-product) of\nR\nand\nS\n. The identity element is the identity relation.\nThe order of\nR\nand\nS\nin the notation\nS ∘ R\n, used here agrees with the standard notational order for [composition of functions](https://bliptext.com/articles/composition-of-functions).\nFor example, the composition \"is mother of\"\n∘\n\"is parent of\" yields \"is maternal grandparent of\", while the composition \"is parent of\"\n∘\n\"is mother of\" yields \"is grandmother of\".\nFor the former case, if\nx\nis the parent of\ny\nand\ny\nis the mother of\nz\n, then\nx\nis the maternal grandparent of\nz\n.\nR\nis a relation over sets\nX\nand\nY\nthen\nRT =\nis the converse relation of\nR\nover\nY\nand\nX\n. For example,\n=\nis the converse of itself, as is\n≠\n, and\n \nare each other's converse, as are\n≤\nand\n≥\n.\nR\nis a relation over\nX\nthen\nR =\n(also denoted by\nR\nor\n¬R\n) is the complementary relation of\nR\n. For example,\n=\nand\n≠\nare each other's complement, as are\n⊆\nand\n⊈\n,\n⊇\nand\n⊉\n, and\n∈\nand\n∉\n, and, for [total orders](https://bliptext.com/articles/total-order), also\n \nand\n≤\n. The complement of the [converse relation](https://bliptext.com/articles/converse-relation)\nRT\nis the converse of the complement:\nR\nis a relation over\nX\nand\nS\nis a subset of\nX\nthen\nRundefined =\nis the of\nR\nto\nS\n. The expression\nRundefined =\nis the of\nR\nto\nS\nRundefined =\nis called the of\nR\nto\nS\n. If a relation is [reflexive](https://bliptext.com/articles/reflexive-relation), ir[reflexive](https://bliptext.com/articles/reflexive-relation), [symmetric](https://bliptext.com/articles/symmetric-relation), anti[symmetric](https://bliptext.com/articles/symmetric-relation), a[symmetric](https://bliptext.com/articles/symmetric-relation), [transitive](https://bliptext.com/articles/transitive-relation), [total](https://bliptext.com/articles/serial-relation), [trichotomous](https://bliptext.com/articles/trichotomy-mathematics), a [partial order](https://bliptext.com/articles/partial-order), [total](https://bliptext.com/articles/serial-relation) order, [strict weak order](https://bliptext.com/articles/strict-weak-order), [total](https://bliptext.com/articles/serial-relation) preorder (weak order), or an [equivalence relation](https://bliptext.com/articles/equivalence-relation), then so too are its restrictions.\nHowever, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal.\nFor example, restricting the relation \"\nx\nis parent of\ny\n\" to females yields the relation \"\nx\nis mother of the woman\ny\n\"; its transitive closure does not 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.\nA relation\nR\nover sets\nX\nand\nY\nis said to be a relation\nS\nover\nX\nand\nY\n, written\nR ⊆ S\n, if\nR\nis a subset of\nS\n, that is, for all\nx ∈ X\nand\ny ∈ Y\n, if\nxRy\n, then\nxSy\n. If\nR\nis contained in\nS\nand\nS\nis contained in\nR\n, then\nR\nand\nS\nare called equal written\nR = S\n. If\nR\nis contained in\nS\nbut\nS\nis not contained in\nR\n, then\nR\nis said to be than\nS\n, written\nR ⊊ S\n. For example, on the [rational numbers](https://bliptext.com/articles/rational-number), the relation\n>\nis smaller than\n≥\n, and equal to the composition\n> ∘ >\n.\n\n# Theorems about relations\nR\nis contained in relation\nS\n, then\nR\nis reflexive, connected, strongly connected, left-total, or right-total, then so is\nS\n.\nS\nis irreflexive, asymmetric, anti-symmetric, left-unique, or right-unique, then so is\nR\n.\n\n# Examples\n\n# Generalizations\nThe above concept of relation has been generalized to admit relations between members of two different sets.\nGiven sets\nX\nand\nY\n, a [heterogeneous relation](https://bliptext.com/articles/heterogeneous-relation)\nR\nover\nX\nand\nY\nis a subset of\n. When\nX = Y\n, the relation concept described above is obtained; it is often called homogeneous relation (or endorelation) to distinguish it from its generalization.\nThe above properties and operations that are marked \"\" and \"\", respectively, generalize to heterogeneous relations.\nAn example of a heterogeneous relation is \"ocean\nx\nborders continent\ny\n\". The best-known examples are [functions](https://bliptext.com/articles/function-mathematics) with distinct domains and ranges, such as\nsqrt : N → R+\n.","revisionCount":0},"hasMore":false,"nextCursor":null}

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