Null String is Identity Element for Concatenation Operator

Theorem

Let $\AA$ be an alphabet of symbols.

Let $\WW$ denote the set of words in $\AA$.

Let $\epsilon$ denote the null string.

Let $C: \WW \times \WW \to \WW$ denote the concatenation operator on $\WW$:

$\forall A, B \in \WW: \map C {A, B} := A B$

Then $\epsilon$ is the identity element for $C$.

As $\epsilon$ is the null string:

$\map C {\epsilon, A}$

$\ds \forall A \in \WW: \, $

$\ds \map C {\epsilon, A}$

$=$

$\ds \epsilon A$

Definition of $C$

$\ds $

$=$

$\ds A$

Definition of Null String

$\ds $

$=$

$\ds A \epsilon$

Definition of Null String

$\ds $

$=$

$\ds \map C {A, \epsilon}$

Definition of $C$

Hence the result by definition of identity element.

$\blacksquare$