Definition:Increasing

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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 φ is increasing iff \forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right).


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 increasing iff x \le y \iff f \left({x}\right) \le f \left({y}\right).


[edit] Sequences

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


Then \left \langle {x_n} \right \rangle is increasing if \forall n \in \N: x_n \le x_{n+1}.

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