Definition: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 decreasing iff:


 * $$\forall x, y \in S: x \preceq_1 y \implies \phi \left({y}\right) \preceq_2 \phi \left({x}\right)$$

Alternative terms are order-inverting, order-reversing, antitone and non-increasing.

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 decreasing iff:
 * $$x \le y \implies f \left({y}\right) \le f \left({x}\right)$$.

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

Then $$\left \langle {x_n} \right \rangle$$ is decreasing iff:
 * $$\forall n \in \N: x_{n+1} \le x_n$$

Also see

 * Strictly decreasing
 * Increasing
 * Monotone