Strictly Increasing Mapping is Increasing

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A mapping that is strictly increasing is an increasing mapping.


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 strictly increasing.

From Strictly Precedes is Strict Ordering:

$x \preceq_1 y \implies x = y \lor x \prec_1 y$


\(\ds x\) \(=\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds \map \phi x\) \(=\) \(\ds \map \phi y\) Definition of Mapping
\(\ds \leadsto \ \ \) \(\ds \map \phi x\) \(\preceq_2\) \(\ds \map \phi y\) as $\preceq_2$, being an ordering, is reflexive

This leaves us with:

\(\ds x\) \(\prec_1\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds \map \phi x\) \(\prec_2\) \(\ds \map \phi y\) Definition of Strictly Increasing Mapping
\(\ds \leadsto \ \ \) \(\ds \map \phi x\) \(\preceq_2\) \(\ds \map \phi y\) Strictly Precedes is Strict Ordering


Also see