I was wondering if I have the relation $R = \{(1, 0), (0, 1) \}$, is there any way I can make the relation antisymmetric without removing elements? I think not, because since the first part of the anti-symmetric implication is true, there is no way the second part $a = b$ will ever be true as long as the initial two elements are part of the relation.
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1No. ${}{}{}{}{}$ – copper.hat Mar 24 '15 at 22:13
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It is possible if you put an equivalence relation $\sim$ on ${0,1}$ such that $0 \sim 1$ and consider $R$ with respect to $\sim$... then $R$ becomes the same as ${(,)}$ on ${*}^2$, though, so it's pretty useless... – A.P. Mar 24 '15 at 22:13
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Thank you, just wanted to make sure, if I understood it correctly – xxsl Mar 24 '15 at 22:13