NationMaster - Encyclopedia: Antisymmetric relation

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

In mathematical notation, this is: Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...

$forall a, b in X, a R b and b R a ; Rightarrow ; a = b$

Inequalities are antisymmetric, since for numbers a and b, a ≤ b and b ≤ a if and only if a = b. The feasible regions of linear programming are defined by a set of inequalities. ...

Note that 'antisymmetric' is not the logical negative of 'symmetric' (whereby aRb implies bRa). (N.B.: Both are properties of relations expressed as universal statements about their members; their logical negations must be existential statements.) Thus, there are relations which are both symmetric and antisymmetric (e.g., the equality relation) and there are relations which are neither symmetric nor antisymmetric (e.g., divisibility on the integers). 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. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... The integers are commonly denoted by the above symbol. ...

Antisymmetry is different from asymmetry. According to one definition of asymmetric, anything that fails to be symmetric is asymmetric; the definition of antisymmetry is more specific than this. Another definition of asymmetric makes asymmetry equivalent to antisymmetry plus irreflexivity. In mathematics, a binary relation R on a set X is antisymmetric if it holds for all a and b in X that if a is related to b and b is related to a then a = b. ... In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ...

Examples

Equality
"... is even, ... is odd"

In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... Image File history File links Evenandodd. ...

Properties containing antisymmetry

Partial order - An antisymmetric relation that is also transitive and reflexive.

total order - An antisymmetric relation that is also transitive and total.

In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, a binary relation R over a set X is total if it holds for all a and b in X that a is related to b or b is related to a (or both). ...

Binary relation - Wikipedia, the free encyclopedia (1260 words)
	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.

Antisymmetric relation - Wikipedia, the free encyclopedia (219 words)
	In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b.
	There are relations which are both symmetric and antisymmetric (equality), relations which are neither symmetric nor antisymmetric (divisibility on the integers), relations which are symmetric and not antisymmetric (congruence modulo n), and relations which are not symmetric but are anti-symmetric ("is less than").
	The relation "is less than or equal to" is not symmetric but is antisymmetric, and the antisymmetric condition is not vacuous.

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Properties containing antisymmetry

See also

Examples