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A relation is antisymmetric if no two distinct elements are mutually related. Together with reflexivity and transitivity it defines a partial order.
$\forall a,b \in X.; (R(a,b) \land R(b,a)) \Rightarrow a = b$
Equivalently: $R \cap R^{-1} \subseteq \Delta_X$
In mathematics, a homogeneous relation R on set X is antisymmetric if there is no pair of distinct elements of X each of which is related by R to the other. More formally, R is antisymmetric precisely if for all a and b in X.
If R(a,b) holds then R(b,a) must not hold
or equivalently: if R(a,b) == R(b,a) then a==b
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