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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 ...
Transitivity is a linguistics property that relates to whether a verb, participle, or gerund denotes a transitive object.It is closely related to valency, which considers other arguments in addition to transitive objects.
An example in modern English is the verb to arrive. Verbs that can be used in an intransitive or transitive way are called ambitransitive verbs. In English, an example is the verb to eat; the sentences You eat (with an intransitive form) and You eat apples (a transitive form that has apples as the object) are both grammatical.
A transitive relation is irreflexive if and only if it is asymmetric. [13] For example, "is ancestor of" is a transitive relation, while "is parent of" is not. Connected for all x, y ∈ X, if x ≠ y then xRy or yRx. For example, on the natural numbers, < is connected, while "is a divisor of " is not (e.g. neither 5R7 nor 7R5). Strongly connected
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".
The reason is that properties defined by bounded formulas are absolute for transitive classes. [3] 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 ...
The concepts of closed sets and closure are often extended to any property of subsets that are stable under intersection; that is, every intersection of subsets that have the property has also the property. For example, in , a Zariski-closed set, also known as an algebraic set, is the set of the common zeros of a family of polynomials, and the ...
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 ...