ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4272 on D5280: Strictly standard-antitone basic real function

Absolute value function is not strictly antitone

Formulation 0

Let $x \mapsto |x|$ be the D412: Absolute value function.

Then

(1)	\begin{equation} 0 < 1 \end{equation}
(2)	\begin{equation} \|0\| < \|1\| \end{equation}

Proofs

Proof 0

Let $x \mapsto |x|$ be the D412: Absolute value function.

The first assertion is clear. By definition of the D412: Absolute value function, both $0$ and $1$ are mapped to themselves (are fixed points). Thus, $|0| = 0 < 1 = |1|$. $\square$