Binary Operation on Subset is Binary Operation

Theorem

Let $S$ be a set.

Let $\circ$ be a binary operation on $S$.

Let $T \subseteq S$.

Let $\circ {\restriction}_T$ be the restriction of $\circ$ to $T$.

Then $\circ {\restriction}_T$ is a binary operation on $T$.

Proof

Let $\Bbb U$ be a universal set.

Let $\circ: S \times S \to \Bbb U$ be a binary operation on $S$.

Let $T \subseteq S$.

Let $\tuple {a, b} \in T \times T$.

By definition of ordered pair and cartesian product:

$a \in T$ and $b \in T$

As $T \subseteq S$, it follows that:

$a \in S$ and $b \in S$

Thus:

$\tuple {a, b} \in S \times S$

As $\circ$ is a binary operation on $S$, it follows that:

$\map \circ {a, b} \in \Bbb U$

But by definition of restriction of $\circ$:

$\map \circ {a, b} = \map {\circ {\restriction}_T} {a, b}$

Thus:

$\map {\circ {\restriction}_T} {a, b} \in \Bbb U$

As $a$ and $b$ are arbitrary elements of $T$, it follows that this holds for all $a, b \in T$.

Hence the result by definition of binary operation.

$\blacksquare$