Relation (mathematics) - Simple English Wikipedia, the free encyclopedia
In mathematics, a function was originally the idealization of how a varying quantity depends on .. This is often used in relation with the arrow notation for elements (read: "f maps x to f (x)"), often stacked immediately below the arrow notation. Mathematical relations fall into various types according to their specific properties , often as expressed in the axioms or definitions that they satisfy. Many of these. In the mathematics of binary relations, the composition relations is a concept of forming a new relation S ∘ R from two given relations R and S. The composition.
The set of all functions is a subset of the set of all relations - a function is a relation where the first value of every tuple is unique through the set. Other well-known relations are the Equivalence relation and the Order relation.
That way, sets of things can be ordered: Take the first element of a set, it is either equal to the element looked for, or there is an order relation that can be used to classify it.
Composition of relations
That way, the whole set can be classified compared to some arbitrarily chosen element. Relations can be transitive.
One example of a transitive relation is "smaller-than". One example of a symmetric relation is "is equal to". If X "is equal to" Y, Y "is equal to" X.
One example of an anti-symmetric relation is if "X is equal Y", "Y is not equal to X". Relations can be reflexive. A relation is asymmetric if and only if it is both anti-symmetric and irreflexive.
For example, "is ancestor of" is transitive, while "is parent of" is not. A transitive relation is irreflexive if and only if it is asymmetric.
This property is sometimes called "total", which is distinct from the definitions of "total" given in the previous section. Every reflexive relation is serial: This makes sense only if relations on proper classes are allowed.
Well-foundedness implies the descending chain condition that is, no infinite chain R x3 R x2 R x1 can exist. If the axiom of choice is assumed, both conditions are equivalent. A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric, transitive, and serial is also reflexive.
Relation - Wikipedia
A relation that is only symmetric and transitive without necessarily being reflexive is called a partial equivalence relation. A relation that is reflexive, antisymmetric, and transitive is called a partial order.
Ancient Greeks[ edit ] Traditionally the history of the concept of relation begins with Aristotle and his concept of relative terms.
In Metaphysics he states: These were later redefined as "categorical" propositions in order to distinguish them from two other types of proposition, the disjunctive and the hypothetical, identified a little later by Chrysippus. A fundamental opposition was developing between substance and relation.
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Basil in the Eastern church suggested that an understanding of the Trinity lay more in understanding the types of relation existing between the three members of the Godhead than in the nature of the Persons themselves.
These were quantity, as in double and half; activity, as in acting and being acted upon; and understanding, through the qualitative concepts of genus and species. But this is plainly seen to be false from the very fact that things themselves have a mutual natural order and relation There are three conditions that make a relation to be real or logical