User:Jshflynn/Empty Word is Two-sided Identity

Theorem

Let $\Sigma$ be an alphabet.

Let $x$ be a word over $\Sigma$, $\lambda$ be the empty word over $\Sigma$ and $\circ$ denote concatenation.

Then $\lambda$ is a two-sided identity element of concatenation. That is,

$x \circ \lambda = x$ and $\lambda \circ x = x$

Proof

Follows immediately from the definition of concatenation with the empty word.

$\blacksquare$