A relation $\mathrel R$ is intransitivity only if $\mathrel R$ is irreflexive. True or False
I think is yes. $\forall x(x\mathrel Rx \wedge x\mathrel Rx) \implies \neg x\mathrel Rx$
Am I correct?
If no , can you explain why?
2026-04-08 05:44:57.1775627097
Question about intransitivity
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You need to distinguish between intransitive and antitransitive. The property you used is antitransitivity which is $\forall a, b, c: (aRb \wedge bRc) \implies \neg aRc$. If you assume antitransitivity, your reasoning is correct, because $xRx$ leads to a contradiction. Intransitivity however is simply the negation of transitivity, which is a weaker property, because it just states that $\exists a,b,c: aRb \wedge bRc \wedge \neg aRc$. From intransitivity you cannot deduce irreflexivity, it is easy to find a counterexample.
See http://en.wikipedia.org/wiki/Intransitivity for more clarification on the terms.