# Definition:Monotone (Order Theory)

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## Definition

### Ordered Sets

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 **monotone** if and only if it is either increasing or decreasing.

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

### Real Functions

This definition continues to hold when $S = T = \R$.

Thus, let $f$ be a real function.

Then $f$ is **monotone** if and only if it is either increasing or decreasing.

### Sequences

Let $\left({S, \preceq}\right)$ be a totally ordered set.

Then a sequence $\left \langle {a_k} \right \rangle_{k \in A}$ of terms of $S$ is **monotone** if it is either increasing or decreasing.

## Notes

This can also be called **monotonic**.