# Definition:Strictly Monotone/Mapping

## Definition

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping.

Then $\phi$ is strictly monotone if and only if it is either strictly increasing or strictly decreasing.

Note that this definition also holds if $S = T$.

## Also known as

This can also be called strictly monotonic.