# Definition:Monotonicity

## 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 monotone mapping.

Then whether $\phi$ is increasing or decreasing is known as the monotonicity of $\phi$.

That is:

if $\phi$ is such that $x \mathop{\preceq_1} y \implies \phi \left({x}\right) \mathop{\preceq_2} \phi \left({y}\right)$, then the monotonicity of $\phi$ is increasing
if $\phi$ is such that $x \mathop{\prec_1} y \implies \phi \left({x}\right) \mathop{\prec_2} \phi \left({y}\right)$, then the monotonicity of $\phi$ is strictly increasing
if $\phi$ is such that $x \mathop{\preceq_1} y \implies \phi \left({y}\right) \mathop{\preceq_2} \phi \left({x}\right)$, then the monotonicity of $\phi$ is decreasing
if $\phi$ is such that $x \mathop{\prec_1} y \implies \phi \left({y}\right) \mathop{\prec_2} \phi \left({x}\right)$, then the monotonicity of $\phi$ is strictly decreasing.