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 $z$.

$\blacksquare$


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