Structure with Element both Identity and Zero has One Element
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $z \in S$ such that $z$ is both an identity element and a zero element.
Then:
- $S = \set z$
Proof
Let $x \in S$.
Then
\(\ds x\) | \(=\) | \(\ds x \circ z\) | Definition of Identity Element | |||||||||||
\(\ds \) | \(=\) | \(\ds z\) | Definition of Zero Element |
and so there is no other element of $S$ but $z$.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $3$