Subtraction has no Identity Element

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Theorem

The operation of subtraction on numbers of any kind has no identity.


Proof

Aiming for a contradiction, suppose there exists an identity $e$ in one of the standard number systems $\GF$.

\(\ds \forall x \in \GF: \, \) \(\ds x\) \(=\) \(\ds x - e\)
\(\ds \) \(=\) \(\ds e - x\)
\(\ds \leadsto \ \ \) \(\ds x + \paren {-e}\) \(=\) \(\ds e + \paren {-x}\)
\(\ds \leadsto \ \ \) \(\ds x + x\) \(=\) \(\ds e + e\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds e\)

That is:

$\forall x \in \GF: x = e$

But from Identity is Unique, if $e$ is an identity then there can be only one such.

From Proof by Contradiction it follows that $\GF$ has no such $e$.

$\blacksquare$


Sources