Restriction of Commutative Operation is Commutative

Theorem

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

Let $T \subseteq S$.

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

Then the restriction $\circ {\restriction_T}$ of $\circ$ to $T$ is also commutative.

$\ds T$

$\subseteq$

$\ds S$

$\ds \leadsto \ \ $

$\ds \forall a, b \in T: \, $

$\ds a, b$

$\in$

$\ds S$

Definition of Subset

$\ds \leadsto \ \ $

$\ds a \mathop {\circ {\restriction_T} } b$

$=$

$\ds a \circ b$

$\ds $

$=$

$\ds b \circ a$

$\circ$ is commutative

$\ds $

$=$

$\ds b \mathop {\circ {\restriction_T} } a$

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.1$. Subsets closed to an operation
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets