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 \phi is increasing iff:

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


Alternative terms are order-preserving, isotone and non-decreasing.


Note that this definition also holds if S = T.


[edit] Note

Some sources insist at the point of definition that \phi be an injection for it to be definable as order-preserving, but this is conceptually unnecessary.


[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 \implies 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 iff:

\forall n \in \N: x_n \le x_{n+1}.


[edit] Also see


[edit] Sources

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