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 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 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 if:
 * $$\forall n \in \N: x_{n+1} < x_n$$.

Also see

 * Decreasing
 * Strictly increasing
 * Strictly monotone