Definition:Strictly Increasing

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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 strictly increasing if and only if:

$\forall x, y \in S: x \prec_1 y \implies \map \phi x \prec_2 \map \phi y$


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:

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


Also see