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From what I am reading, antisymmetric means:

$$∀ x ∀ y \,[ R ( x , y ) ∧ R ( y , x ) ⇒ x = y ]$$

However, $(2,1)$ and $(1,2)$, $X\ne Y$. I understand how this is symmetric but how is this antisymmetric? I got this from my professor and my book explains that they are not mutually exclusive.

  • $\begingroup$ Closely related is this question. $\endgroup$ – Cameron Buie May 16 '15 at 18:21
  • $\begingroup$ Much belated: it is not antisymmetric, for precisely the reason you state. We have $(1,2)$ and $(2,1)$ related, but $1\neq 2.$ $\endgroup$ – Cameron Buie Mar 7 at 13:53

Consider matrix which has ones on diagonal and zeros on other places. Symmetric property: $\forall a,b\in X$ $aRb\implies bRa$. Antisymmetric property: $\forall a,b\in X$ ($aRb \land bRa)\implies a=b$. So consider relation $R=\{(x_1,x_1),(x_2,x_2)...(x_n,x_n)\}$ s.t. $x_i\in X$ As you see both properties are hold, so we get matrix - $a_{ij}=1$ for $i=j$ and $a_{ij}=0$ for $i\neq j$.


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