Restriction of Antisymmetric Relation is Antisymmetric

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:

\(\ds \set {\tuple {x, y}, \tuple {y, x} }\)

\(\subseteq\)

\(\ds \RR {\restriction_T}\)

\(\ds \leadsto \ \ \)

\(\ds \set {\tuple {x, y}, \tuple {y, x} }\)

\(\subseteq\)

\(\ds \paren {T \times T} \cap \RR\)

Definition of Restriction of Relation

\(\ds \leadsto \ \ \)

\(\ds \set {\tuple {x, y}, \tuple {y, x} }\)

\(\subseteq\)

\(\ds \RR\)

Intersection is Subset

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds y\)

$\RR$ is Antisymmetric on $S$

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

$\blacksquare$

Also see

• Properties of Restriction of Relation‎ for other similar properties of the restriction of a relation.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings