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	PlanetMath: symmetric relation
		that is both symmetric and antisymmetric has the property that
		This is version 17 of symmetric relation, born on 2002-02-02, modified 2006-10-19.
		Object id is 1647, canonical name is Symmetric.
	planetmath.org /encyclopedia/Symmetric.html (42 words)

	Symmetric relation - Wikipedia, the free encyclopedia
		In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.
		There are relations which are both symmetric and antisymmetric (equality), there are relations which are neither symmetric nor antisymmetric (divisibility), there are relations which are symmetric and not antisymmetric (congruence modulo n), and there are relations which are not symmetric but are anti-symmetric ("is less than or equal to").
		A symmetric relation that is also transitive and reflexive is an equivalence relation.
	en.wikipedia.org /wiki/Symmetric_relation (152 words)

	Binary relation - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Binary_relation (1262 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)

	Math Forum - Ask Dr. Math (Site not responding. Last check: )
		We can think of that relation as "mapping" 1 to both 1 and 2 and also mapping 5 to 4; thus we see that it is all right to map an element to more than one element (1 was mapped to 1 and to 2).
		First, it is fine to say that R is symmetric because in the statement of the problem, we are told that we are dealing with a symmetric relation.
		A relation R is symmetric if for all (x,y) in R, we have that (y,x) is also in R. Now remember that R and R^n are different sets.
	mathforum.org /library/drmath/view/51847.html (2012 words)

	plope - Descriptions of Relation Terminology (Site not responding. Last check: )
		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)

	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 sister of" is not symmetric on a set that contains a brother and sister but would be symmetric on a set of females.
		The relation "is an ancestor of" on a set of people is transitive as is the empty relation on a set.
	www.math.csusb.edu /notes/rel/node2.html (334 words)

	Binary relation
		A binary relation is a mathematical concept to do with "relations", such as "is greater than" and "is equal to" in arithmetic, or "is an element of" in set theory.
		Thus the first element of R is the set of objects, the second is the set of people, and the last element is a set of ordered pairs of the form (object, owner).
		If a binary relation is also a binary function injective and onto, the converse is called inverse of the function.
	www.brainyencyclopedia.com /encyclopedia/b/bi/binary_relation.html (924 words)

	[No title]
		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)

	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)

	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.
		What it basically says is that a relation is anti-symmetric if the relation is symmetric only when x and y that are the same element.
	www.cs.uni.edu /~schafer/courses/080/sessions/s12.htm (808 words)

	ON VARIOUS TYPES OF RELATIONS (Site not responding. Last check: )
		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)

	Physics Help and Math Help - Physics Forums - Equivalence Relations
		Equivalence relations are the generalization of the relation of being equal - whilst two elements are not the same, they may in share all the properties that you care about.
		I just thought of a binary relation that is symmetric but not reflexive and not transitive: the relation "is perpendicular to" on the set of vectors in a real three-dimensional vector space.
		The binary relation "divides," on the set of positive integers (as in "5 divides 15, 5 does not divide 12") is reflexive and transitive, but not symmetric.
	www.physicsforums.com /printthread.php?t=15405&pp=40 (1107 words)

	[No title] (Site not responding. Last check: )
		Since R is symmetric (2) and (3) imply xRa (4) and bRx (5).
		R(S is a binary relation from A to C. Then (R(S)-1 is a binary relation from C to A: (R(S)-1 = {(c, a)
		Assume R is symmetric to prove that R= s(R)=R(R-1(1).
	longwood.cs.ucf.edu /courses/cot3100.spr2000/honor/lect14.doc (1297 words)

	CmSc 180 – Discrete mathematics (Site not responding. Last check: )
		"less than" is not a symmetric relation, it is anti-symmetric.
		Equality is a transitive relation: a = b, b = c, hence a = c
		is the relation symmetric, anti-symmetric, or neither symmetric nor anti-symmetric
	www.simpson.edu /~sinapova/cmsc180a/L26-properties.htm (644 words)

	No Title
		The parent of relations, ``...is a parent of...'', is a binary relation between pairs of people.
		In a symmetric relation, the matrix is symmetric around the main diagonal.
		This relation is reflexive, symmetric, and transitive, and hence is an equivalence relation.
	www.cs.sunysb.edu /~skiena/113/lectures/lecture22/lecture22.html (923 words)

	Math 161 Homework 2 (Site not responding. Last check: )
		Remark 2: Of course R is a subset of S, so (i) and (ii) mean that (in a sense) S is the smallest symmetric relation that contains R as a subset.
		Suppose F is a nonempty set of symmetric relations.
		Suppose R is a relation on a set A. Prove there is an equivalence relation E on A such that R is a subset of E. (There may be many such E's.)
	math.stanford.edu /~white/161_w05/161hw2.htm (570 words)

	Relations
		When defining a relation, you can also define an inverse relation, which is a relation between a relation target (the second class) and a relation source (the first class).
		A symmetrical relation creates its own inverse relation, which has the same name as the relation, as described in Defining a Symmetric Relation.
		The inverse of a relation has the inverse cardinality of that relation, as this table shows.
	www.cs.fsu.edu /g2/g2doc/g2rm/relatio6.htm (174 words)

	Relations (Site not responding. Last check: )
		This means that when you create a relation between two items, by default, G2 creates a single relation between the relation source and the relation target.
		If you define a relation to be symmetrical, however, concluding an instance of that relation also concludes an inverse relation of the same name as the relation.
		is both the relation name and the inverse relation name of the symmetric relation.
	www.cs.fsu.edu /g2/g2doc/g2rm/relatio7.htm (206 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).
		is a relation R on a set S where R is reflexive, symmetric, and transitive.
	www.lv.psu.edu /OJJ/courses/ist-230/topics/relations.html (659 words)

	Partial Order Scoring (statistics)
		Note that the commonly used 'equality' relation, (=), defined on the set of real numbers is an equivalence relation.
		In the situation of equality, the comparison is continued using variables of the next priority level.
		This procedure is repeated until the relation is determined at one of the priority levels, or the end of the variable list is reached.
	www.unesco.org /webworld/idams/Doc/ManualHtml/E2poscor.htm (499 words)

	Reversible Unification Grammars
		In sign-based approaches such as UCG [55] and HPSG [34], the string of words is not assigned a privileged status but is represented as the value of one of the attributes of a feature structure.
		It is also possible to use unification grammars to define other symmetric relations between feature structures.
		Each unification grammar defines a symmetric relation, for example between Dutch strings and Dutch logical forms, or between Dutch logical forms and English logical forms.
	odur.let.rug.nl /vannoord/papers/austin/node9.html (312 words)

	relation3
		This symbol represents a binary function whose first argument is a set S, whose second argument is a relation R on S. When applied to S and R, it represents the smallest symmetric relation (with respect to inclusion) on S containing R. Signatures:
		This symbol represents the boolean binary function which returns true if and only if the second argument is a symmetric relation on the first.
		The classes of a reflexive relation R on S cover S, as a in S belongs to class(S,R,a).
	www.win.tue.nl /~amc/oz/om/cds/relation3.html (902 words)

	[No title] (Site not responding. Last check: )
		A least element but A greatest element but no greatest element no least element¡(WZª ÎóRO¨Summary¨,A binary relation on a set S is a subset of SxS.
		Binary relations can have properties of reflexivity, symmetry, anti-symmetry, and transitivity.
		A equivalence relation on a set S defines a partition of S. Partial orders.
	www.cst.cmich.edu /users/manou1a/175/175-relation.ppt (187 words)

	[No title]
		Definition: xRx is true for all x in the set Examples: Equality is a reflexive relation for any object x: x = x is true.
		"less then" (defined on the set of real numbers) is not a reflexive relation.
		-@-À-Â-N...0/r/t/v/x/¦/¨/ýýýýýýýùýýýýýýýôýôìýôìýýýýý Ð¤x &F are in relation R, then y and x are also in R, i.e. if xRy is true, yRx is also true. 1. Examples: equality is a symmetric relation: if a = b then b = a "less than" is not a symmetric relation : if a
	www.simpson.edu /~sinapova/cmsc180/cmsc180-04/L24-Properties.doc (145 words)

	The Analytic/Synthetic Distinction
		In all judgments in which the relation of a subject to the predicate is thought (if I only consider affirmative judgments, since the application to negative ones is easy) this relation is possible in two different ways.
		Either the predicate B belongs to the subject A as something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it.
		Katz (1972) (in arguments that can be regarded as independent of the appeals to intuition we considered in §4.1) draws attention to related semantic data, such as subjects' agreements about, e.g., synonymy, redundancy, antonomy, and implication, as well as to what he believes are the serious prospects of systematically relating syntactic and semantic structure.
	plato.stanford.edu /entries/analytic-synthetic (8176 words)

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