Definition:Decreasing

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[edit] 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, antitone and non-increasing.


Note that this definition also holds if S = T.


[edit] 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).


[edit] 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


[edit] Also see


[edit] Sources

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