# Idempotent element

## Definition 2

An element $$x$$ of a set $$S$$ is idempotent under an operation $$\circ$$ if and only if:

$$x \circ x = x$$
THEOREM

Zero and one are the only idempotent elements of natural number multiplication.

\begin{align} & 0 * 0 = 0 \\ & 1 * 1 = 1 \\ & (x + 2) * (x + 2) \ne x + 2 \end{align}

Proof unavailable

THEOREM

Zero is the only idempotent element of natural number addition.

\begin{align} & 0 + 0 = 0 \\ & (x + 1) + (x + 1) \ne x + 1 \end{align}

Proof unavailable