# Definition:Strictly Increasing

## 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 strictly increasing if:

$\forall x, y \in S: x \prec_1 y \implies \phi \left({x}\right) \prec_2 \phi \left({y}\right)$

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

### Real Functions

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

Let $f$ be a real function.

Then $f$ is strictly increasing if and only if:

$x < y \implies f \left({x}\right) < f \left({y}\right)$

### Sequences

Let $\struct {S, \preceq}$ be a totally ordered set.

Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is strictly increasing if and only if:

$\forall j, k \in A: j < k \implies a_j \prec a_k$