User:Jshflynn/Empty Word is Unique

Theorem

The empty word over an alphabet is unique.

Proof

Let $\Sigma$ be an alphabet.

The empty word is defined to be any word of length $0$.

Supposing there is more than one empty word and we choose any two of them labelling them $\lambda$ and $\lambda '$.

Then by assumption:

$\operatorname{len}(\lambda) = \operatorname{len}(\lambda ') = 0$ .

And:

$\forall i: \lambda_i = \lambda'_i$ (Holds vacuously)

So by the definition of word equality:

$\lambda = \lambda'$.

Hence there is only one empty word over $\Sigma$

$\blacksquare$