Right Cancellable Element is Right Cancellable in Subset
Jump to navigation
Jump to search
Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $\struct {T, \circ} \subseteq \struct {S, \circ}$.
Let $x \in T$ be right cancellable in $S$.
Then $x$ is also right cancellable in $T$.
Proof
Let $x \in T$ be right cancellable in $S$.
That is:
- $\forall a, b \in S: a \circ x = b \circ x \implies a = b$
Therefore:
- $\forall c, d \in T: c \circ x = d \circ x \implies c = d$
Thus $x$ is right cancellable in $T$.
$\blacksquare$