Right Cancellable Commutative Operation is Left Cancellable

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Theorem

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

Let $\circ$ be right cancellable and also commutative.


Then $\circ$ is also left cancellable.


Proof

Let $\circ$ be both right cancellable and commutative on a set $S$.

Then:

\(\ds a \circ b\) \(=\) \(\ds a \circ c\)
\(\ds \leadsto \ \ \) \(\ds b \circ a\) \(=\) \(\ds c \circ a\) $\circ$ is Commutative
\(\ds \leadsto \ \ \) \(\ds b\) \(=\) \(\ds c\) $\circ$ is Right Cancellable

$\blacksquare$


Also see