Cancellation Laws/Proof 1

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Let $G$ be a group.

Let $a, b, c \in G$.

Then the following hold:

Right cancellation law
$b a = c a \implies b = c$
Left cancellation law
$a b = a c \implies b = c$


Let $a, b, c \in G$ and let $a^{-1}$ be the inverse of $a$.

Suppose $b a = c a$.


\(\ds \paren {b a} a^{-1}\) \(=\) \(\ds \paren {c a} a^{-1}\)
\(\ds \leadsto \ \ \) \(\ds b \paren {a a^{-1} }\) \(=\) \(\ds c \paren {a a^{-1} }\) Definition of Associative Operation
\(\ds \leadsto \ \ \) \(\ds b e\) \(=\) \(\ds c e\) Definition of Inverse Element
\(\ds \leadsto \ \ \) \(\ds b\) \(=\) \(\ds c\) Definition of Identity Element

Thus, the right cancellation law holds.

The proof of the left cancellation law is analogous.