Like, if $(a,b)$ belongs to $R$ and $(b,c)$ belongs to $R$, then if $(a,c)$ also belongs to $R$, it is a transitive relation. Can we take $(b,b)$ in place of $(b,c)$ so that it comes out $(a,b)$ belongs to $R$, and $(b,b)$ belongs to $R$, which means $(a,b)$ belongs to $R$ if the relation is transitive. Is this valid?. Can we take $b=c $ to prove transitivity? Help is appreciated.
How to prove if a relation is transitive?
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elementary-set-theory
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0Why did I get downvoted? – 2017-01-14
1 Answers
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A relation $R$ defined in $E$ is transitive $\iff$
$$\forall (a,b,c)\in E^3$$
$a $ R $ b$ and $b $ R $ c \implies a$ R $ c.$
to prove no transitivity, you need find $(u,v,w)\in E^3\;\;:\;$
$u $ R $v \;, v$ R $w $ but $ u \not $ R $ w$.