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Transitive definition is $\forall x[(x,y)∈R \land (y,z)∈R \implies (x,z)∈ R]$.

If $(a,b), (b,a) ∈ R$, then can I say that $(a,a) ∈ R$?

Because based on the definition $x,y,z$ are not the same value. So, I am not sure whether $(a,a)$ can be a transitive.

hardmath
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SinLok
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1 Answers1

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The definition of transitive for relations applies to all $x,y,z$ in the domain of the relation. It does not require them to be distinct. So if $R$ is a transitive relation, then the reasoning $(a,b),(b,a) \in R$ implies $(a,a) \in R$ is valid.

Note however that a relation $R$ being transitive and symmetric does not necessarily imply that $R$ is reflexive, the sort of thing your argument seems intended to show. The gap in such a proof is that for some $a$ in the domain of $R$, there may not be any pair $(a,b)\in R$ and so the argument fails to get the premise it needs to start with.

hardmath
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  • I have a exercise question and solution like this: The set {1,2,3,4} Show reflexive, symmetric, and not transitive. Then, the solution is R={(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (2,3), (3,2)} Why (1,1) and (1,2) => (1,2) is not transitive? – SinLok Dec 11 '16 at 13:29
  • In your case the failure of transitivity is not just about $(1,2)$ belonging to the relation, but rather about the pairs $(1,2)$ and $(2,3)$ belonging to the relation. If it were transitive, what would the implication be? – hardmath Dec 11 '16 at 13:39
  • (1,2) and (2,3) => should be include (1,3) for transitive. But the question ask to show Not transitive. So that, I don't understand why the solution also include (1,2). It the solution wrong? – SinLok Dec 11 '16 at 13:45
  • It seems to be a wrong solution for that part of the problem. Maybe it applies to some other part? When asking about a textbook exercise it is often helpful to give Readers the author and title of the textbook and where the exercise is found. Note that the relation you gave is not transitive because $(1,3)$ is not in the relation. – hardmath Dec 11 '16 at 13:49
  • This is a individual question. Maybe my school give me a wrong solution. Thank you for helping me! – SinLok Dec 11 '16 at 13:57