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We have a relation R = {$(a,a),(b,b)$}.

Is this relation transitive? If that is true, then why is it transitive? According to definition a relation is transitive

If (a,b) in R & (b,c) in R then (a,c) in R

$\forall a,b,c \in R: a R b \land b R c \implies a R c$.

But in our set we're missing the element c so how could it be transitive?

KReiser
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MoveUK
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  • So neither (a,b) not (b,c) are in the relation. The antecedent is false. The implication is true. There is no conflict with the relation being transitive. – Graham Kemp Nov 02 '18 at 00:33
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    PS: Don't confuse the use of letters as variables in definitions with the use of letters as values in the problem – Graham Kemp Nov 02 '18 at 00:43

2 Answers2

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The relation will not be transitive only if there is a counterexample - where $(x,y)$ and $(y,z)$ are related but $(x,z)$ are not.

There we have no such counterexample example.

Note: Don't confuse $a,b,c$ being used as variables in a definition, with $\rm a,b,c$ being used as enumerator values in the sets.

Graham Kemp
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It is definitely a TRANSITIVE RELATION. see for transitive we say IF ( IF word is important ) if (a,b) belongs to a set , and (b,c) belongs to set ,then ( a,c) must be there for the set to be transitive. OR if these type of order pair are absent( like in your question) it is definitely transitive