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Topic: Irreflexive relation

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	Binary relation - Wikipedia, the free encyclopedia
		Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic, "is congruent to" in geometry, "is adjacent to" in graph theory, and many more.
		A binary relation that is functional is called a partial function; a binary relation that is both left-total and functional is called a function.
		Certain mathematical "relations", such as "equal to", "member of", and "subset 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.
	en.wikipedia.org /wiki/Binary_relation (1898 words)

	Encyclopedia :: encyclopedia : Binary relation (Site not responding. Last check: 2007-10-25)
		In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as "is greater than" and "is equal to" in arithmetic, or "is congruent to" in geometry, or "is an element of" or "is a subset of" in set theory.
		Put in lay terms, a binary relation is a statement about two objects that may be true or false depending on the choice of objects, for example, "4 is less than 5" is true, and the relation is "is less than".
		A binary relation that is functional is called a partial function; a binary relation that is both total and functional is called a function.
	www.hallencyclopedia.com /Binary_relation (1255 words)

	Reflexive relation - Wikipedia, the free encyclopedia
		An irreflexive (or aliorelative) relation R is one where for all a in X, a is never R-related to itself.
		Irreflexivity is a stronger condition than failure of reflexivity, so a binary relation may be reflexive, irreflexive, or neither.
		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.
	en.wikipedia.org /wiki/Reflexive_relation (201 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-25)
		Reflexive relations ------------------- A relation is reflexive if it contains all the pairs (x,x) for every x in A. In the example above, this means that we must have '' in all squares of the main diagonal - the smallest possible reflexive relation* is:
		The number of irreflexive relations is therefore the same as the number of reflexive relations.
		Note that "irreflexive" is not the same as "not reflexive." "Irreflexive" means you have no squares on the diagonal; "not reflexive" means you don't have all the squares on the diagonal.
	mathforum.org /library/drmath/view/62565.html (1616 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.
		Example 5: The relation = on the set of integers {1, 2, 3} is {<1, 1>, <2, 2> <3, 3> } and it is symmetric.
		(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)

	Hierarchy - Facts, Information, and Encyclopedia Reference article
		A hierarchy is a transitive, irreflexive, asymmetric relationship, such as "is superior to", "is part of", or "is taller than":
		When two nodes are related, one is designated the "superior" (or sometimes the "parent") and the other the "subordinate" (or sometimes the "child").
		In the intuitive case of the "is the boss of" relation, the boss is the superior and the employee is the subordinate.
	www.startsurfing.com /encyclopedia/b/e/a/Bearer.html (1321 words)

	lec11Sept
		A relation is antisymmetric if there is never an edge from both a to b and b to a, when a and b are different.
		A relation is asymmetric if there is never an edge from both a to b and from b to a, and there are no loops.
		A relation is transitive if whenever there is an edge from a to b and from b to c, there is an edge completing the triangle by going from a to c directly.
	www.pitt.edu /~vanlehn/cs0441/lec21Oct.html (861 words)

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

	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)

	Download Info of - Binary relation (Site not responding. Last check: 2007-10-25)
		In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set (mathematics) with elements of another (perhaps the same) set.
		An example is the "divides" relation between the set of prime number s P and the set of integer s Z, in which every prime p is associated to every integer z that is a divisibility of p.
		A binary relation that is functional is called a partial function ; a binary relation that is both left-total and functional is called a function (mathematics).
	binary.relation.en.cwap.org (3077 words)

	[No title]
		The relation “is not equal to” is a symmetric relation.
		The relation “is married to” is a symmetric relation.
		The relation “is an offspring of” is a transitive relation.
	www.weizmann.ac.il /usersfiles/zemed/upload/(FILE)1142323994.doc (1846 words)

	Topic 1
		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)

	LECTURE
		A relation may be neither reflexive nor irreflexive.
		It means that a relation is irreflexive if in its matrix representation the diagonal elements are all zero, if one of them is not zero the we will say that the relation is not irreflexive.
		Let R be an anti-symmetric relation on a set A. Then by definition, no two distinct elements of A are related to each other.
	zahid11.20m.com /lec13.htm (957 words)

	Binary_relation
		The order of the elements in each pair of G is important: if a ≠ b, then a R b and b R a can be true or false, independently of each other.
		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.
		Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation which is a set.
	www.brainyencyclopedia.com /encyclopedia/b/bi/binary_relation.html (1930 words)

	Equivalence Relation
		When a relation is derived from a set crossed with itself, several important properties become meaningful.
		A relation possessing all three properties is called an equivalence relation.
		The relation r is a partial ordering on the set s, or s is a partially ordered set via r, or s is a poset, if r is transitive and antisymmetric.
	www.mathreference.com /set,rst.html (1118 words)

	CmSc 180 – Discrete mathematics (Site not responding. Last check: 2007-10-25)
		Thus R2 is irreflexive, while R3 is neither reflexive nor irreflexive.
		"less than" is not a symmetric relation, it is anti-symmetric.
		Equality is a transitive relation: a = b, b = c, hence a = c
	storm.simpson.edu /~sinapova/cmsc180a/L26-properties.htm (644 words)

	BinaryRelation
		reflexive relation irreflexive relation symmetric relation antisymmetric relation trichotomizing relation transitive relation intransitive relation unary function binary predicate
		relation is disjointly decomposed into binary relation,ternary relation,quaternary relation,quintary relation,variable arity relation.
		If is an instance of binary relation, then there don't exist,,, so that (,,,) holds.
	godel.iis.sinica.edu.tw /kifb/en/concepts/_binary_relation.html (313 words)

	No Title
		The parent of relations, ``...is a parent of...'', is a binary relation between pairs of people.
		This relation is reflexive, symmetric, and transitive, and hence is an equivalence relation.
		What an equivalence relation means is that the elements can be partitioned into equalence classes (or blocks) such that all elements in the same block have the same properties.
	www.cs.sunysb.edu /~skiena/113/lectures/lecture22/lecture22.html (923 words)

	PartWhole - ESW Wiki (Site not responding. Last check: 2007-10-25)
		The partOf relation is one of the basic structuring primitives of the universe.
		The overlaps relation is typically used to express the constraint of supplementation, that is, that two things that overlap must have at least one part that they do not share (otherwise they are the same).
		A common confusion is with the constitution relation, which is a relation between things and the material or substance from which they are made.
	esw.w3.org /topic/PartWhole (517 words)

	IrreflexiveRelation
		Relation rel is irreflexive if (rel inst inst) holds for no value of inst.
		antisymmetric relation binary predicate intransitive relation reflexive relation symmetric relation transitive relation trichotomizing relation unary function
		If is an instance of irreflexive relation, then for all holds: (,) doesn't hold.
	godel.iis.sinica.edu.tw /kifb/en/concepts/_irreflexive_relation.html (107 words)

	[No title]
		The main principles concerning these relations can be formulated within the framework of classical extensional mereology supplemented with some specific axioms on the behavior of the hole-in relation".) ;; Primitive predicates ;; ;; 2-place: part-of (documentation "This section of the theory introduces the 2-place 'part-of' relation.
		The relation is taken to satisfy the clause that what is connected with a part is also connected with the whole, so that if ?x is a part of ?y, then everything that ?x is connected with ?y is also connected with.
		This too is a monotonic relation, in the sense that if ?x is partially filled by ?y and ?y is part of ?z, then ?x is partially filled by ?z.
	suo.ieee.org /SUO/ontologies/Holes.txt (2186 words)

	Relations
		is a relation R on a set S where R is reflexive, antisymmetric, and transitive, i.e.
		The relation of divisibility is a partial order on the set S = Div (n).
		, then the relation "a is congruent to b (mod n)" is an equivalence relation.
	www.lv.psu.edu /OJJ/courses/ist-230/topics/relations.html (659 words)

	[No title]
		~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Constraints ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Constraint relation: A relation on Real^n.
		An element of the relation is said to "fulfill" the constraint.
		Projection of a relation to the i-th component: The set of the projections of all elements of R to the i-th component.
	www.mpi-sb.mpg.de /~suchanek/personal/texts/summaries/cr.txt (9802 words)

	Relation Practice Exercises (Site not responding. Last check: 2007-10-25)
		Draw the digraph for the following relation R on the set A={1,2,3,4,5,6}: aRb if and only if a is a factor (i.e., divisor) of b.
		Consider the relation R={ (1,2), (1,3), (3,1), (1,1), (3,3), (3,2), (1,4), (4,2), (3,4) }.
		Explain why an asymmetric relation must be irreflexive.
	home.snu.edu /~kwantz/m1113/relationpractice.htm (102 words)

	[No title]
		The ordering of the fields is unimportant, except that consistency must be maintained; that is, each column in the table contains values for a specific field in all of the tuples.
		The number of tuples (rows) is called the cardinality of the relation.
		This subset is called the primary key of the relation.
	www.cs.umb.edu /~khsuyan/note/Session20.ppt (499 words)

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