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Topic: Reflexive relation

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Transitive relation

Equivalence relation

Equality (mathematics)

Empty set

Cartesian product

Modular arithmetic

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	Reflexive relation - Wikipedia, the free encyclopedia
		A reflexive relation R on set X is one where for all a in X, a is R-related to itself.
		The strict inequalities "less than" and "greater than" are irreflexive relations whereas the inequalities "less than or equal to" and "greater than or equal to" are reflexive.
		However, if we define a relation R on the integers such that a R b iff a = -b, then it is neither reflexive nor irreflexive, because 0 is related to itself.
	en.wikipedia.org /wiki/Reflexive_relation (205 words)

	Equivalence Relation
		Equivalence relation, a mathematical concept, is a type of relation on a given set that provides a way for elements of that set to be identified with (meaning considered equivalent to for some present purpose) other elements of that set.
		Equivalence relation is defined in a branch of mathematics called set theory, a vital branch underpinning all branches of mathematics and those fields that use mathematics.
		Equivalence relations are so ubiquitous in mathematics and other fields that use mathematics because they enable the user to partition a set in a particular way of the user’s design.
	www.iscid.org /encyclopedia/Equivalence_Relation (2634 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-26)
		Reflexive Relation From the definition we know a relation is reflexive if every element of it is related with itself.
		The relation {(1,1)} is symmetric, since (x,y) in R is only true for x = y = 1, and, in that case, we also have (y,x) in R (it's the same pair).
		Consider now the relation: R = {(1,2),(2,1)} In this case, if we let x = 1, y = 2, z = 1, we see that (x,y) = (1,2) is in R (y,z) = (2,1) is in R (x,z) = (1,1) is not in R and this shows that the relation is not transitive.
	mathforum.org /library/drmath/view/63001.html (1421 words)

	Course Summary
		Reflexive property: (a,a) is an element of R for every a in A. Symmetric property: if (a,b) is an element of R, (b,a) is an element of R. Antisymmetric property: if (a,b) is an element of R and (b,a) is an element of R, then a=b.
		An n-ary relation is a subset of A
		The reflexive closure is the union of the relation with the identity relation - the relation in which (a,a) is always an element of the relation.
	sweb.uky.edu /~jcscov0/dmath_4.htm (1397 words)

	Order Relation
		Example 4: The relation {< 1, 1 >, < 1, 2 >, < 1, 3 >, < 2, 3>, < 3, 3 > } on the set of integers {1, 2, 3} is neither reflexive nor irreflexive.
		(a) is reflexive, antisymmetric, symmetric and transitive, but not irreflexive.
		(b) is neither reflexive nor irreflexive, and it is antisymmetric, symmetric and transitive.
	www.cs.odu.edu /~toida/nerzic/content/relation/property/property.html (550 words)

	Relations (Related to Ch
		The composite of R and S is the relation consisting of ordered pairs (a,c), where a is from A and c is in C, and for which there exists an element b in B such that aRb and bSc.
		Suppose R is a relation from A to B and S is a relation from B to C, where A= {1, 2,3}, B = { 1, 2,3, 4}, and C = {0, 1,2} with R = { (1, 1), (1,4), (2, 3), (3, 1), (3, 4)} and S = { (1,0), (2,0), (3,1), (3,2), (4,1)}.
		Let R be an equivalence relation on a set A. The set of all elements that are related to an element a of A is called the equivalence class of a.
	cms.dt.uh.edu /faculty/delavinae/Math_2305/Relations/Relation.html (1047 words)

	Reflexive - Wikipedia, the free encyclopedia
		reflexive pronoun (grammar), a pronoun with a reflexive relationship with its self-identical antecedent
		reflexive verb (grammar), where a semantic agent and patient are the same
		reflexive relation (mathematics), a certain relation where elements of a set are self-related
	en.wikipedia.org /wiki/Reflexive (173 words)

	lec11Sept
		A relation from set A to set B is a set of ordered pairs (x,y) such that xÎA and yÎB.
		A symmetric relation on A: "xÎA "yÎA [ (x,y)ÎR -> (y,x)ÎR ]
		An asymmetric relation on A: "xÎA "yÎA [ (x,y)ÎR -> (y,x)ÏR.]
	www.pitt.edu /~vanlehn/cs0441/lec16Oct.html (254 words)

	Relations on a set
		An example of a non reflexive relation is the relation "is the father of" on a set of people.
		The relation "is the father of " is irreflexive.
		The relation "is the sister of" is not symmetric on a set that contains a brother and sister but would be symmetric on a set of females.
	www.math.csusb.edu /notes/rel/node2.html (334 words)

	PlanetMath: reflexive
		would be a reflexive relation, because it contains all the
		This is version 8 of reflexive, born on 2002-02-02, modified 2004-05-29.
		Object id is 1644, canonical name is Reflexive.
	planetmath.org /encyclopedia/Reflexive.html (51 words)

	plope - Descriptions of Relation Terminology
		A relation R is non-transitive iff it is neither transitive nor intransitive.
		A relation R is non-symmetric iff it is neither symmetric nor asymmetric.
		A relation R is non-reflexive iff it is neither reflexive nor irreflexive.
	www.plope.com /Members/chrism/relationship_terminology (471 words)

	Rational Mathematics: Relations
		Thus a relation R can be also represented in truth-table form, where the rows are labelled with the elements of X and the columns with the elements of Y and the table entries are 1 or 0 according as (a, b) are related by R or not.
		Simple relations can often helpfully be represented by network diagrams, in which the elements of XuY are denoted by marked and labelled points, and the pair (x, y) in R is denoted by an arrowed line joining point x to point y.
		For an equivalence relation to partition a set A it must be reflexive in A or, equivalently, every element of A must be in the domain of the equivalence relation.
	homepage.ntlworld.com /gpjnow/RM2-relations.htm (976 words)

	[No title]
		A binary relation from A to B is a subset of A X B (A cross B).
		The relation R is antisymmetric iff (a, b) (R and (b, a) (R imply that a = b, i.e.
		Suppose that R is a relation from set A to set B, and S is a relation from set B to set C. Also suppose that A, B, and C have m, p, and n elements respectively.
	www.cecs.csulb.edu /~pompei/cecs228/ch6.doc (2745 words)

	Unit 6 Notes
		This relation is symmetric, but not reflexive: (1,1) is not in the relation, and not transitive: (1,3) and (3,1) are in the relation, but not (1,1), for example.
		This relation is not reflexive: (b,b) not in R, not symmetric: (b,a) not in R, but it is transitive: the only pair (x,y) in R, (y,z) in R where x not equal to y and y not equal to z is (a,b) and (b,c), but (a,c) is in R. (d)
		For a binary relation R to be not transitive we need only have that there exists some a, b, c such that a R b, b R c, and a not R c.
	www-cse.ucsd.edu /groups/Gill-rawfiles/SolWeb_CSE20/u6notes.html (3175 words)

	Relations
		Thus the intersection of a transitive relation with its reverse is trivially transitive.
		The reverse of a right-reflexive relation is trivially left-reflexive; and vice-versa.
		The reverse of a reflexive relation is trivially reflexive.
	www.chaos.org.uk /~eddy/maths/found/relate.html (2539 words)

	[Coq-Club] problem setoid
		Relation rel -> Relation (unify_relation_carrier_with_type env rel t) (* first order matching with a bit of conversion ) ( Note: the type checking operations performed by the function could ) ( be done once and for all abstracting the morphism structure using ) ( the quantifiers.
		Relation rel -> match rel.rel_refl with None -> errorlabstrm "Setoid_reflexivity" (str "The relation " ++ prrelation rel ++ str " is not reflexive.")
		Relation rel -> match rel.rel_sym with None -> errorlabstrm "Setoid_symmetry" (str "The relation " ++ prrelation rel ++ str " is not symmetric.")
	pauillac.inria.fr /pipermail/coq-club/2005/001690.html (5696 words)

	CmSc 365 Theory of Computation (Site not responding. Last check: 2007-10-26)
		The reflexive closure of the relation R is obtained by the union of R and the identity relation I
		Composition of two binary relations is computed as boolean matrix multiplication of the binary matrices that represent the relations.
		The transitive closure is equal to the original relation, hence the original relation R is transitive.
	storm.simpson.edu /~sinapova/cmsc365-02/L03-Closures.htm (1012 words)

	Computers and Vision Lecture - 29 August 1994 (Site not responding. Last check: 2007-10-26)
		It is often helpful to envision a relation as a directed graph with nodes corresponding to the elements of A and arcs corresponding to the pairs in R.
		One forms the reflexive closure of a non-reflexive relation R on set A by adding to R the pair (a,a) for each a in A.
		The reflexive-transitive closure of a relation is the result of forming the transitive closure of the reflexive closure of the relation.
	www.cise.ufl.edu /~jnw/VisionCourse/Lectures/94.08.29.html (465 words)

	New Page 1
		A good example of this is the "sum is an odd number" relation (ok, we haven't looked at that yet, but consider that if x+y is an odd number then clearly y+x is an odd number so this relation would be symmetric).
		This property basically says that if two elements have a relation, and two other elements have the same relation, and the first element of one is the same as the second element of the other then the first element of the other has the relation with the second element of the one.
		To make sure that the relation is NOT AS, we must have at least one situation where (x,y) and (y,x) are in the relation set and x≠y.
	www.cs.uni.edu /~schafer/courses/080/sessions/s12.htm (808 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		EQUIVALENCE CLASSES By definition, (from your text), if R is an equivalence relation on a set S, then the equivalence class of an element a (in set S), denoted [a], is the set of all elements in S which are related to a.
		We could choose the relation R = {(a,a),(b,c),(c,b)}, and we could graph it like this: a --- a b --- c c --- b A relation which pairs only one second element with any first element is known as a FUNCTION.
		Relations which are not functions: R = {(1,2), (1,3), (2,3), (3,4)} R = {(a,a), (b,b), (c,c), (c,d)} Relations which are functions: R = {(1,1),(2,2),(3,3)} R = {(a,b),(b,c),(c,d)} Up until now, we have been looking at the relation as a set containing the ordered pairs which are in the relation.
	www.unf.edu /public/cot3100/jgiles/lecture6 (1722 words)

	GAP Manual: 88 Binary Relations
		Each relation of degree n is considered an element of the full relation monoid of degree n although it is not necessary to construct a full relation monoid before working with relations.
		Then there are a function which tests whether an arbitrary object is a relation (see IsRelation) and a function which determines the degree of a relation (see Degree of a Relation).
		A relation R subseteq {1, dots, n} times {1, dots, n} is transitive if (x, z) in R whenever (x, y) in R and (y, z) in R for some y in {1, dots, n}.
	schmidt.nuigalway.ie /gap/CHAP088.htm (2211 words)

	Discrete mathematics:Functions and relations - Wikibooks, collection of open-content textbooks
		is a relation and not a function, since both 1 and 2 are mapped to two values, 1 and -1, and 2 and -2 respectively.
		This is a relation (not a function) since we can observe that 1 maps to 2 and 3, for instance.
		It is true that when we are dealing with relations, we may find that many values are related to one fixed value.
	en.wikibooks.org /wiki/Discrete_mathematics:Functions_and_relations (2333 words)

	CmSc 180 – Discrete mathematics (Site not responding. Last check: 2007-10-26)
		"less than" is not a symmetric relation, it is anti-symmetric.
		is the relation reflexive, irreflexive, or neither reflexive nor irreflexive
		is the relation symmetric, anti-symmetric, or neither symmetric nor anti-symmetric
	www.simpson.edu /~sinapova/cmsc180a/L26-properties.htm (644 words)

	Topic 1 (Site not responding. Last check: 2007-10-26)
		The relation of being in love with is not irreflexive; it is only not reflexive.
		"Being an ancestor of" is a transitive relation.
		In math this relation is often spoken of as equality.
	www.thelogiccourse.com /bluestorm/9h14.html (800 words)

	No Title (Site not responding. Last check: 2007-10-26)
		The parent of relations, ``x is a parent of y'', is a binary relation between pairs of people.
		Is the boss-of relation reflexive, irreflexive, or neither?
		This relation is reflexive, symmetric, and transitive, and hence is an equivalence relation.
	www.cs.sunysb.edu /~cse113/ref/lecture22/lecture22.html (978 words)

	Exercise 2.7 - Answers
		To put it another way: if a relation is reflexive, and the domain is not empty, there are loops in its graph; but loops are double arrows; so it is not asymmetric.
		This is possible, as long as the relation is empty and the domain has just one member - otherwise, because it is connected, the relation can't be empty.
		An example would be a relation whose graph contained two dots with a loop on one and a single arrow from one to the other.
	logic.philosophy.ox.ac.uk /exAnswers/ex8-1answer.htm (710 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		REFLEXIVE PROPERTY For the relation to be reflexive: A x A a b c For the set A, ALL of these must be in R: -------------- a
		SYMMETRIC PROPERTY For the relation to be symmetric: A x A a b c For the set A, "mirror image" pairs must be present.
		Since R is reflexive, symmetric, and transitive, we say that R is an EQUIVALENCE relation.
	www.unf.edu /public/cot3100/jgiles/lecture5 (1387 words)

	ON VARIOUS TYPES OF RELATIONS
		In a relational database, a table corresponding to a reflexive relation must contain 2 rows for each pair that satisfies the relation.
		The 2 approaches (devising a check to enforce a row for every permutation or tagging the relation as 'symmetric') discussed so far for the symmetric 2-place relation are not attractive for the general case of n-place.
		The usual arguments against complex types discourage this, I think you agree. One is almost tempted to amend the relational model to allow sets of columns in a table to be designated as unordered with respect to each other so that each row in the table enumerates a set.
	www.dbdebunk.com /page/page/622108.htm (798 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		Any reflexive relation is a subset of this set of 12 elements; we know there exist 212 such subsets.
		A relation R on a set A is called irreflexive if for all a(A, (a, a)(R. Give an example of a relation R on Z where R is irreflexive, and transitive but not symmetric.
		Assume that R is nonempty relation that is irreflexive, symmetric and transitive.
	longwood.cs.ucf.edu /courses/cot3100.spr2001/homework/hmk3_key.doc (794 words)

	M3000 Homework #13
		and we have that n R p, i.e., the relation is transitive.
		Therefore, (x,y) R (u,v) and the relation is transitive.
		Since 3 does not divide 2, 1 S 1 is not true, showing that the relation is not reflexive.
	www-math.cudenver.edu /~wcherowi/courses/m3000/abhw13.html (1404 words)

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