Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

b ( i. e., take the reflexive closure of the relation ).

This gives the partial order associated with the strict partial order "<" through reflexive closure ; in this case the equivalence is equality, so we don't need the notations and ~.

Here represents the reflexive and transitive closure of the step relation meaning any number of consecutive steps ( zero, one or more ).

The reflexive reduction of a binary relation ~ on a set S is the smallest relation ~′ such that ~′ shares the same reflexive closure as ~.

It can be seen in a way as the opposite of the reflexive closure.

This particular reflex is simply one of several aero digestive reflexes such as the reflexive pharyngeal swallow, the pharyngoglottal closure reflex, in which no swallowing occurs yet the glottis still closes, and the pharyngo-upper esophageal sphincter contractile reflex, occurring mainly during gastroesophageal reflux episodes.

The semantics for the common knowledge operator, then, is given by taking, for each group of agents G, the reflexive and transitive closure of the, for all agents i in G, call such a relation, and stipulating that is true at state s iff is true at all states t such that.

A zero-or-more-steps rewriting like this is captured by the reflexive transitive closure of, denoted by ( see abstract rewriting system # Basic notions ).

Similarly, the reflexive transitive symmetric closure of, denoted ( see abstract rewriting system # Basic notions ), is a congruence, meaning it is an equivalence relation ( by definition ) and it is also compatible with string concatenation.

In mathematics, a directed set ( or a directed preorder or a filtered set ) is a nonempty set A together with a reflexive and transitive binary relation ≤ ( that is, a preorder ), with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.

A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive.

In mathematics, especially in order theory, a preorder or quasi-order is a binary relation that is reflexive and transitive.

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i. e., for all a, b and c in P, we have that:

The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary relations which are reflexive, symmetric, and transitive.

In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i. e., a relation ~ on S where x ~ x holds true for every x in S. For example, ~ could be " is equal to ".

An irreflexive, or anti-reflexive, relation is the opposite of a reflexive relation: it is a binary relation on a set where no element is related to itself.

For example, the binary relation " the product of x and y is even " is reflexive on the set of even numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of natural numbers.

A set with a binary relation R on its elements that is reflexive ( for all a in the set, aRa ), antisymmetric ( if aRb and bRa, then a = b ) and transitive ( if aRb and bRc, then aRc ) is described as a partially ordered set or poset.

A well-quasi-ordering on a set is a quasi-ordering ( i. e., a reflexive, transitive binary relation ) such that any infinite sequence of elements,,, … from contains an increasing pair ≤ with <.

If we assume that the binary relationship is also reflexive, then it follows that thermal equilibrium is an equivalence relation.

This relation is clearly symmetric and transitive, but in view of the existence of odd numbers, it is not reflexive.

* The relation "≥" between real numbers is reflexive and transitive, but not symmetric.

* The relation " has a common factor greater than 1 with " between natural numbers greater than 1, is reflexive and symmetric, but not transitive.

* The empty relation R on a non-empty set X ( i. e. aRb is never true ) is vacuously symmetric and transitive, but not reflexive.

* The relation " is approximately equal to " between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change.

Although siblinghood is symmetric ( if A is a sibling of B, then B is a sibling of A ) and transitive on any 3 distinct people ( if A is a sibling of B and C is a sibling of B, then A is a sibling of C, provided A is not C ( Note that " is a sibling of " is NOT a transitive relation, since A R B, and B R A implies A R A by transitivity )), it is not reflexive ( A cannot be a sibling of A ).

* A partial order is a relation that is reflexive, antisymmetric, and transitive.

Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric.

* A reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite.

If a relation is Euclidean and reflexive, it is also symmetric and transitive.

Hence an equivalence relation is a relation that is Euclidean and reflexive.

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder ( weak order ), an equivalence relation, or a relation with any other special properties, if and only if R is.

( The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preorder twice, and symmetric by definition.

Transitive and symmetric imply reflexive if and only if for all a ∈ X, there exists a b ∈ X such that a ~ b.

Nevertheless, key aspects of feminist theorizing and methods became de rigueur as part of the ' post-modern moment ' in anthropology: Ethnographies became more reflexive, explicitly addressing the author's methodology, cultural, gender and racial positioning, and their influence on his or her ethnographic analysis.

* Lemma A Banach space X is reflexive if and only if the natural pairing on X × X ′ is perfect.

This relies on the reflexive response due to " overlearning " the skill of morse code reception / detection / transcription so that it is an autonomous function requiring no specific attention to perform.

While reflexive existence is not considered by materialists to be experienced on the atomic level, the individual's physical and mental experiences are ultimately reducible to the unique tripartite combination of environmentally determined, genetically determined, and randomly determined interactions of firing neurons and atomic collisions.

Operant behavior operates on the environment and is maintained by its consequences, while classical conditioning deals with the conditioning of reflexive ( reflex ) behaviors which are elicited by antecedent conditions.

In all cases it should be reflexive, make a substantial contribution toward the understanding of the social life of humans, have an aesthetic impact on the reader, and express a credible reality.

Later " reflexive " ethnographies refined the technique to translate cultural differences by representing their effects on the ethnographer.

He believed that for the most part, Christianity had forsaken its mystical tradition in favor of Cartesian emphasis on “ the reification of concepts, idolization of the reflexive consciousness, flight from being into verbalism, mathematics, and rationalization.

According to it, the contemporary form of science's existing autonomy is the reflexive autonomy: actors and structures within the scientific field are able to translate or to reflect diverse themes presented by social and political fields, as well as influence them regarding the thematic choices on research projects.