Definition:Increasing
Jump to navigation
Jump to search
Definition
Ordered Sets
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: S \to T$ be a mapping.
Then $\phi$ is increasing if and only if:
- $\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$
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 increasing if and only if:
- $x \le y \implies \map f x \le \map f y$
Sequences
Let $\struct {S, \preceq}$ be a totally ordered set.
Let $A$ be a subset of the natural numbers $\N$.
Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is increasing if and only if:
- $\forall j, k \in A: j < k \implies a_j \preceq a_k$