# Restriction of Associative Operation is Associative

## Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $T \subseteq S$.

Let $\circ_T$ denote the restriction of $\circ$ to $T$.

Let the operation $\circ$ be associative on $\struct {S, \circ}$.

Then $\circ_T$ is associative on $\struct {T, \circ_T}$.

## Proof

 $\ds T$ $\subseteq$ $\ds S$ $\ds \leadsto \ \$ $\ds \forall a, b, c \in T: \,$ $\ds a, b, c$ $\in$ $\ds S$ Definition of Subset $\ds \leadsto \ \$ $\ds a \mathop {\circ_T} \paren {b \mathop {\circ_T} c}$ $=$ $\ds a \circ \paren {b \circ c}$ $\ds$ $=$ $\ds \paren {a \circ b} \circ c$ as $\circ$ is associative $\ds$ $=$ $\ds \paren {a \mathop {\circ_T} b} \mathop {\circ_T} c$

$\blacksquare$