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A relation is symmetric if when aRb then bRa, antisymmetric if when aRb and bRa then a=b, and asymmetric if when aRb then not bRa.

Does it follow that every antisymmetric relation R is also reflexive (Id is a subset of R), and that a relation both antisymmetric and symmetric is equal to the identity relation (Id = R)?

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You're correct about all of the definitions, but your conclusions are not true. Let $A$ be our set, and take $R$ to be any subset of $\Delta_A = \{(a,a) \mid a\in A\}$. This is a relation which is both symmetric and antisymmetric, but is not reflexive, since there's no reason to suggest that it is all of $\Delta_A$.

As a concrete example, take $A=\{1,2,3\}$ and $R=\{(1,1)\}$. It is clearly symmetric, and it trivially satisfies anti-symmetry.

Hayden
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  • Thank you for your help, but I'm still missing something. Like, the concrete example, I can see it's not reflexive because (a,a) is an element of R only for a = 1, but using this logic wouldn't it also not be anti-/symmetric, as (a,b) element of R is not satisfied for every a,b of A? Or I'm completetly lost. –  Dec 04 '16 at 19:10
  • The definition of anti-symmetry says that whenever we have $a,b$ such that $aRb$ and $bRa$, then $a=b$. In our case, if we have such a pair $a,b$ with $aRb$ and $bRa$, then it must be that $(a,b)=(1,1)$, since this is the only element in $R$. Thus, $a=b$. – Hayden Dec 04 '16 at 19:12
  • Okay thanks, this definition thing sure goes deep. Is it the secret to math wizardry? ;) Thanks for helping me sort things out! –  Dec 04 '16 at 19:25
  • That's the benefit of formality and rigor in math. Sometimes a lot of these things can be confusing without precise wording. If you're familiar with quantifiers and writing out logical formula, that can sometimes be helpful. Otherwise, try to be as rigorous as possible when you're writing a definition so you know exactly what it's saying – Hayden Dec 04 '16 at 19:29
  • Okay, I'll keep that in mind. Regarding logic - do you have any recommendations I should be looking into? –  Dec 04 '16 at 23:30