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An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y". The transitive closure of this relation is "some day x comes after a day y on the calendar", which is trivially true for all days of the week x and y (and thus equivalent to the Cartesian square , which is " x and y are both ...
The semantics for the common knowledge operator, then, is given by taking, for each group of agents G, the reflexive (modal axiom T) and transitive closure (modal axiom 4) 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 (,).
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 ...
Various inequivalent definitions of Kleene algebras and related structures have been given in the literature. [5] Here we will give the definition that seems to be the most common nowadays. A Kleene algebra is a set A together with two binary operations + : A × A → A and · : A × A → A and one function * : A → A , written as a + b , ab ...
In matroid theory, the closure of X is the largest superset of X that has the same rank as X. The transitive closure of a set. [1] The algebraic closure of a field. [2] The integral closure of an integral domain in a field that contains it. The radical of an ideal in a commutative ring.
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A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system (provided that the element relation of the model is the restriction of the true element relation to the universe of the model). Transitivity is an important factor in determining the absoluteness of formulas.
English: The transitive closure of a directed acyclic graph. The original graph is shown by the heavier blue edges. The original graph is shown by the heavier blue edges. The red edges, added to form the transitive closure, connect pairs of reachable vertices: the first vertex of each red edge can reach the second one by a path in the blue graph.