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In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x ...
The original motivation for introducing semiorders was to model human preferences without assuming that incomparability is a transitive relation.For instance, suppose that , , and represent three quantities of the same material, and that is larger than by the smallest amount that is perceptible as a difference, while is halfway between the two of them.
Relational frame theory (RFT) is a behavioral theory of human language. It is rooted in functional contextualism and focused on predicting and influencing verbal behavior with precision, scope and depth. [8] Relational framing is relational responding based on arbitrarily applicable relations and arbitrary stimulus functions.
A further distinction is between transitive and intransitive relations. Transitive relations exhibit a chain-like nature: if x is related to y and y is related to z, then x is related to z. An example is the relation being larger than: if a truck is larger than a car and a car is larger than a bicycle then a truck is larger than a bicycle. A ...
The transitive closure of this relation is a different relation, namely "there is a sequence of direct flights that begins at city x and ends at city y". Every relation can be extended in a similar way to a transitive relation. An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y".
Transitive set. Class of mathematical set whose elements are all subsets. In set theory, a branch of mathematics, a set is called transitive if either of the following equivalent conditions holds: whenever , and , then . whenever , and is not an urelement, then is a subset of . Similarly, a class is transitive if every element of is a subset of .
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, " is less than " is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values ...
For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle ...