Subset Product with Identity
(Redirected from Subset Product by Identity Singleton)
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Theorem
Let $\struct {S, \circ}$ be a magma.
Let $\struct {S, \circ}$ have an identity element $e$.
Then $e \circ S = S \circ e = S$, where $\circ$ is understood to be the subset product with singleton.
Proof
\(\ds e \circ S\) | \(=\) | \(\ds \set e \circ S\) | Definition of Subset Product with Singleton | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {x \circ y: x \in \set e, \, y \in S}\) | Definition of Subset Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {e \circ y: y \in S}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {y: y \in S}\) | Definition of Identity Element | |||||||||||
\(\ds \) | \(=\) | \(\ds S\) |
Thus:
- $e \circ S = S$
A similar argument shows that:
- $S \circ e = S$
$\blacksquare$