सकर्मक संबंध

Transitive relation

Type	Binary relation
Field	Elementary algebra
Statement	A 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 $c$.
Symbolic statement	$\forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc$

गणित में, समुच्चय पर संबंध | संबंध R सेट पर Xसकर्मक है अगर, सभी तत्वों के लिए a, b, c में X, जब भी R संबंधित a को b और b को c, तब R संबंध भी रखता है a को c. प्रत्येक आंशिक क्रम के साथ-साथ प्रत्येक तुल्यता संबंध को सकर्मक होना चाहिए।

परिभाषा

Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation File:Green check.svgY ✗ ✗ ✗ ✗ ✗ File:Green check.svgY ✗ ✗ Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ File:Green check.svgY ✗ ✗ Partial order ✗ File:Green check.svgY ✗ ✗ ✗ ✗ File:Green check.svgY ✗ ✗ Total preorder ✗ ✗ File:Green check.svgY ✗ ✗ ✗ File:Green check.svgY ✗ ✗ Total order ✗ File:Green check.svgY File:Green check.svgY ✗ ✗ ✗ File:Green check.svgY ✗ ✗ Prewellordering ✗ ✗ File:Green check.svgY File:Green check.svgY ✗ ✗ File:Green check.svgY ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ File:Green check.svgY ✗ ✗ File:Green check.svgY ✗ ✗ Well-ordering ✗ File:Green check.svgY File:Green check.svgY File:Green check.svgY ✗ ✗ File:Green check.svgY ✗ ✗ Lattice ✗ File:Green check.svgY ✗ ✗ File:Green check.svgY File:Green check.svgY File:Green check.svgY ✗ ✗ Join-semilattice ✗ File:Green check.svgY ✗ ✗ File:Green check.svgY ✗ File:Green check.svgY ✗ ✗ Meet-semilattice ✗ File:Green check.svgY ✗ ✗ ✗ File:Green check.svgY File:Green check.svgY ✗ ✗ Strict partial order ✗ ✗ ✗ ✗ ✗ ✗ ✗ File:Green check.svgY File:Green check.svgY Strict weak order ✗ ✗ ✗ ✗ ✗ ✗ ✗ File:Green check.svgY File:Green check.svgY Strict total order ✗ ✗ File:Green check.svgY ✗ ✗ ✗ ✗ File:Green check.svgY File:Green check.svgY Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all $a,b$ and $S\neq \varnothing :$ ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$
File:Green check.svgY indicates that the column's property is required by the definition of the row's term (at the very left). For example, the definition of an equivalence relation requires it to be symmetric. ✗ indicates that the property may, or may not hold. All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc,$ and there are additional properties that a homogeneous relation may satisfy.

एक सजातीय संबंध R मंच पर X सकर्मक संबंध है यदि,

सबके लिए a, b, c ∈ X, यदि a R b और b R c, तब a R c.

या पहले क्रम के तर्क के संदर्भ में:

$\mathrm{\forall <!--\; \forall \; -->}a,b,c\in <!--\; \in \; -->X:(a$