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[ Relations / Set Theory ] Is it true that all relations both symmetric and antisymmetric are transitive?
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VinceMiguel is
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What I've done until now: If R is symmetric, then we have that xRy -> yRx, and, if R is antisymmetric, then we have that (xRy and yRx) -> x==y. Now, thanks to symmetry, we know that (xRy and yRx) <-> xRy, correct ? Going back to the definition of antisymmetry, we now have that xRy -> x==y. So our relation is the relation of equality (=), which is indeed transitive.
Does this conclude that all symmetric and antisymmetric relations are transitive? Thanks in advance!
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