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There are three properties of relation,

 1. Reflexive
 2. Symmetric 
 3. Transitive

and if all properties are satisfy by a relation then its known as Equivalence. Now if the relation is reflexive + anti-symmetric + transition then its known as Partial Order Relation.. My question is that is there a relation like, when irreflexive + anti-symmetric + anti-transitive (don't know if its exist, means the negation of transitive) then what we call it ?

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    $\begingroup$ No there is no such thing as anti-transitive. You might define it as $xRy\land yRz\Rightarrow\lnot xRz$, but it wouldn't be a very interesting property. The negation of transitive $xRy\land yRz\not\Rightarrow xRz$, which just says the relation is not transitive, would be even less interesting. Note that for instance anti-symmetric is not the negation of symmetric either. $\endgroup$
    – Marc van Leeuwen
    Commented Nov 29, 2014 at 8:45
  • $\begingroup$ Thanks for helping me .. @Marc van Leeuwen $\endgroup$
    – anaszaman
    Commented Nov 29, 2014 at 8:57

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No there is no such thing as anti-transitive. You might define it as xRy∧yRz⇒¬xRz, but it wouldn't be a very interesting property. The negation of transitive xRy∧yRz⇏xRz, which just says the relation is not transitive, would be even less interesting. Note that for instance anti-symmetric is not the negation of symmetric either (By Marc van Leeuwen)

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