Restriction of Antisymmetric Relation is Antisymmetric

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Theorem

Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be an antisymmetric relation on $S$.


Let $T \subseteq S$ be a subset of $S$.

Let $\mathcal R {\restriction_T} \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$.


Then $\mathcal R {\restriction_T}$ is an antisymmetric relation on $T$.


Proof

Suppose $\mathcal R$ is antisymmetric on $S$.

Then:

\(\displaystyle \left\{ {\left({x, y}\right), \left({y, x}\right)}\right\}\) \(\subseteq\) \(\displaystyle \mathcal R {\restriction_T}\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left\{ {\left({x, y}\right), \left({y, x}\right)}\right\}\) \(\subseteq\) \(\displaystyle \left({T \times T}\right) \cap \mathcal R\) Definition of Restriction of Relation
\(\displaystyle \implies \ \ \) \(\displaystyle \left\{ {\left({x, y}\right), \left({y, x}\right)}\right\}\) \(\subseteq\) \(\displaystyle \mathcal R\) Intersection is Subset
\(\displaystyle \implies \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle y\) $\mathcal R$ is Antisymmetric on $S$

Thus $\mathcal R {\restriction_T}$ is antisymmetric on $T$.

$\blacksquare$


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