Definition:Strictly Decreasing

Ordered Sets
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping.

Then $\phi$ is strictly decreasing (or strictly order-reversing) if:


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

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 strictly decreasing (or strictly order-reversing) iff:
 * $x < y \iff f \left({y}\right) < f \left({x}\right)$

Sequences
Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Then $\left \langle {x_n} \right \rangle$ is strictly decreasing (or strictly order-reversing) if:
 * $\forall n \in \N: x_{n+1} < x_n$

Also see

 * Decreasing
 * Strictly increasing
 * Strictly monotone