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
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.