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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 ...
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.
However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother of the woman y"; its transitive closure does not relate a woman with her paternal grandmother. On the other ...
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 ...
Many languages, including English, have ditransitive verbs that denote two objects, and some verbs may be ambitransitive in a manner that is either transitive (e.g., "I read the book" or "We won the game") or intransitive (e.g., "I read until bedtime" or "We won") depending on the given context.
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.
For example, restricting the relation "is parent of " to females yields the relation "is mother of the woman "; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.
The closure of V is the direct sum of all powers of V. = = Suppose M is a set and A is the set of all binary relations on M. Taking + to be the union, · to be the composition and * to be the reflexive transitive closure, we obtain a Kleene algebra.