Consider a set A = {0,1,2,3}, a relation R from set A = {(0,0),(1,1),(2,2),(3,3)}. Is relation R transitive? If yes, how?

discrete-mathematics elementary-set-theory relations

I was told relation R is transitive but for it to be transitive, (a,b) and (b,c) must belong to R, and only then if (a,c) belongs to R, it is transitive, so how is it transitive?


Solution 1:

Suppose $a,b,c\in A$
we will show $aRb$ and $bRc \implies aRc$
$a=b=c$
$a=c$
$aRc$
so R is transitive.

the subset of $A\times A$ for which the relation $R$ holds is the diagonal of $A\times A$.

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