Restriction of Antisymmetric Relation is Antisymmetric

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Theorem

Let $S$ be a set.

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


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

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


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


Proof

Suppose $\RR$ is antisymmetric on $S$.

Then:

\(\displaystyle \set {\tuple {x, y}, \tuple {y, x} }\) \(\subseteq\) \(\displaystyle \RR {\restriction_T}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \set {\tuple {x, y}, \tuple {y, x} }\) \(\subseteq\) \(\displaystyle \paren {T \times T} \cap \RR\) Definition of Restriction of Relation
\(\displaystyle \leadsto \ \ \) \(\displaystyle \set {\tuple {x, y}, \tuple {y, x} }\) \(\subseteq\) \(\displaystyle \RR\) Intersection is Subset
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle y\) $\RR$ is Antisymmetric on $S$

Thus $\RR {\restriction_T}$ is antisymmetric on $T$.

$\blacksquare$


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