Why isn't reflexivity repetitive in the definition of equivalence relationship?

An equivalence relationship is specified by 3 buildings: reflexivity, proportion and also transitivity.

Does not proportion and also transitivity indicates reflexivity? Take into consideration the adhering to argument.

For any kind of $a$ and also $b$, $a R b$ indicates $b R a$ by proportion. Making use of transitivity, we have $a R a$.

Resource: Workout 8.46, P195 of Mathematical Proofs, 2nd (not 3rd) ed. by Chartrand et alia

2019-05-04 17:36:24
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Answers: 2

Actually, without the reflexivity problem, the vacant relationship would certainly count as an equivalence relationship, which is non - perfect.

Your argument made use of the theory that for each and every $a$, there exists $b$ such that $aRb$ holds. If this holds true, after that proportion and also transitivity indicate reflexivity, yet this is not real as a whole.

2019-05-08 08:33:39


The missing out on problem is occasionally called 'seriality' - - for any kind of x there have to be a y such that x R y.

If you add seriality to the proportion and also transitivity you get a reflexive relationship once more.

2019-05-08 08:26:26