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Intransitivity
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===Properties=== * An antitransitive relation is always [[Irreflexive relation|irreflexive]]. * An antitransitive relation on a set of β₯4 elements is never [[Connex relation|connex]]. On a 3-element set, the depicted cycle has both properties. * An irreflexive and [[Left-unique relation|left-]] (or [[Right-unique relation|right-]]) unique relation is always anti-transitive.<ref>If ''aRb'', ''bRc'', and ''aRc'' would hold for some ''a'', ''b'', ''c'', then {{math|1=''a'' = ''b''}} by left uniqueness, contradicting ''aRb'' by irreflexivity.</ref> An example of the former is the ''mother'' relation. If ''A'' is the mother of ''B'', and ''B'' the mother of ''C'', then ''A'' cannot be the mother of ''C''. * If a relation ''R'' is antitransitive, so is each subset of ''R''.
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