ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Binary cartesian set product
▼ Binary relation
▼ Binary endorelation
Definition D289
Antisymmetric binary relation
Formulation 0
A D4424: Binary endorelation $B = (X \times X, R)$ is antisymmetric if and only if \begin{equation} \forall \, x, y \in X \left( (x, y), (y, x) \in R \quad \implies \quad x = y \right) \end{equation}
Formulation 1
A D4424: Binary endorelation $B = (X \times X, R)$ is antisymmetric if and only if \begin{equation} \forall \, x, y \in X \left( x R y, \, y R x \quad \implies \quad x = y \right) \end{equation}
Remarks
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Remark 0 (Proof technique: establishing an equality by applying antisymmetry)
Let $R$ be an [[[d,289]]] on $X \neq \emptyset$ and let $x, y \in X$ each be a [[[d,2218]]] in $X$. If one was interested in establishing the equality $x = y$ then, due to antisymmetry, it is sufficient to prove that $(x, y) \in R$ and $(y, x) \in R$ since this would imply $x = y$.