Search results
Results from the WOW.Com Content Network
The converse is not true: most directed graphs are neither reflexive nor transitive. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph.
A relation is transitive if it is closed under this operation, and the transitive closure of a relation is its closure under this operation. A preorder is a relation that is reflective and transitive. It follows that the reflexive transitive closure of a relation is the smallest preorder containing it. Similarly, the reflexive transitive ...
In mathematics, the transitive closure R + of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive.For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets R + is the unique minimal transitive superset of R.
In fact, even some relations that are not partial orders are of special interest. Mainly the concept of a preorder has to be mentioned. A preorder is a relation that is reflexive and transitive, but not necessarily antisymmetric. Each preorder induces an equivalence relation between elements, where a is equivalent to b, if a ≤ b and b ≤ a ...
A transitive relation is ... A strict partial order is also known as an asymmetric strict preorder. ... via the operations of reflexive closure (cls ...
The transitive extension of R 1 would be denoted by R 2, and continuing in this way, in general, the transitive extension of R i would be R i + 1. The transitive closure of R, denoted by R* or R ∞ is the set union of R, R 1, R 2, ... . [8] The transitive closure of a relation is a transitive relation. [8]
Transitive closure, R + Defined as the smallest transitive relation over X containing R. This can be seen to be equal to the intersection of all transitive relations containing R. Reflexive transitive closure, R* Defined as R* = (R +) =, the smallest preorder containing R. Reflexive transitive symmetric closure, R ≡
Since it is the union of all simulations, it is the unique largest simulation. Simulations are also closed under reflexive and transitive closure; therefore, the largest simulation must be reflexive and transitive. From this follows that the largest simulation—the simulation preorder—is indeed a preorder relation. [1]