Inverses of Elements Related by Compatible Relation/Corollary
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Corollary to Inverses of Elements Related by Compatible Relation
Let $\struct {G, \circ}$ be a group with identity $e$.
Let $\RR$ be a relation compatible with $\circ$.
Then the following hold:
- $\forall x, y \in G:$
- $x \mathrel \RR e \iff e \mathrel \RR x^{-1}$
- $e \mathrel \RR x \iff x^{-1} \mathrel \RR e$
Proof
From Inverse of Identity Element is Itself:
- $e^{-1} = e$
From Inverses of Elements Related by Compatible Relation:
- $\forall x, y \in G: x \mathrel \RR y \iff y^{-1} \mathrel \RR x^{-1}$
Substituting $e$ for $y$ gives:
- $x \mathrel \RR e \iff e \mathrel \RR x^{-1}$
Substituting $e$ for $x$ and $x$ for $y$ gives:
- $e \mathrel \RR x \iff x^{-1} \mathrel \RR e$
$\blacksquare$