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If a relation is transitive then its transitive extension is itself, that is, if R is a transitive relation then R 1 = R. 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 ...
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 ...
All definitions tacitly require the homogeneous relation be transitive: for all ,,, if and then . A term's definition may require additional properties that are not listed in this table. In mathematics , a binary relation associates elements of one set called the domain with elements of another set called the codomain . [ 1 ]
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 .
Reflexive and transitive: The relation ≤ on N. Or any preorder; Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0. Or any partial equivalence relation; Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3." Or any dependency relation.
All definitions tacitly require the homogeneous relation be transitive: for all ,,, if and then . A term's definition may require additional properties that are not listed in this table. In mathematics , a binary relation R {\displaystyle R} on a set X {\displaystyle X} is reflexive if it relates every element of X {\displaystyle X} to itself.
In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. That is, we can find three values a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} where the transitive condition does not hold.
All definitions tacitly require the homogeneous relation be transitive: for all ,,, if and then . A term's definition may require additional properties that are not listed in this table. A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order .