Cancellation Laws/Corollary 1
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Corollary to Cancellation Laws
Let $g$ and $h$ be elements of a group $G$ whose identity element is $e$.
Then:
- $g h = g \implies h = e$
Proof 1
\(\ds g h\) | \(=\) | \(\ds g\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds g h\) | \(=\) | \(\ds g e\) | Group Axiom $\text G 2$: Existence of Identity Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds h\) | \(=\) | \(\ds e\) | Left Cancellation Law |
$\blacksquare$
Proof 2
\(\ds g h\) | \(=\) | \(\ds g\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds g^{-1} \paren {g h}\) | \(=\) | \(\ds g^{-1} g\) | Group Axiom $\text G 2$: Existence of Identity Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {g^{-1} g} h\) | \(=\) | \(\ds g^{-1} g\) | Group Axiom $\text G 1$: Associativity | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds e h\) | \(=\) | \(\ds e\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds h\) | \(=\) | \(\ds e\) | Group Axiom $\text G 2$: Existence of Identity Element |
$\blacksquare$