# Definition:Monotone (Order Theory)

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## Definition

### Ordered Sets

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ 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 $\struct {S, \preceq}$ be a totally ordered set.

A sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is **monotone** if and only if it is either increasing or decreasing.

## Notes

This can also be called **monotonic**.