"Antisymmetric relation is an equivalence relation" Whenever I used to prove this statement I come with a counterexample. Let take A= {1,2,3,4} If R= {(1,1),(2,3),(3,4)} then R is antisymmetric but not equivalent. Can someone explain to me how the above statement is true?
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4Where is this coming from? The statement is obviously wrong. – mathcounterexamples.net Sep 11 '21 at 14:07
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In general, it is not true that every antisymmetric relation is an equivalent relation but we can have antisymmetric relations which are also equivalent relations.
For example $$ R=\{(1,1), (2,2), (3,3)\} $$is both equivalent and antisymmetric on the set $ A= \{1,2,3\}.$
Mohammad Riazi-Kermani
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