Monotone function

English

A nondecreasing monotone function

A nonincreasing monotone function

Alternative forms

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Noun

1. Template:Lb A function f : X→R (where X is a subset of R, possibly a discrete set) that either never decreases or never increases as its independent variable increases; that is, either x ≤ y implies f(x) ≤ f(y) or x ≤ y implies f(y) ≤ f(x). Template:C

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2. Template:Lb A function f : X→Y (where X and Y are posets with partial order "≤") with either: (1) the property that x ≤ y implies f(x) ≤ f(y), or (2) the property that x ≤ y implies f(y) ≤ f(x).
3. Template:Lb A Boolean function with the property that switching any one input variable from 0 to 1 results either in no change in output or a change from 0 to 1.

Usage notes

• The order theory definition avoids reference to the concepts Template:M and Template:M, making it somewhat more generally applicable. Strictly speaking, the partial orders for X and Y need not be related (the notation "≤" is conventional). This case encompasses the possibility that X and Y are multidimensional spaces (e.g. Rⁿ) and f is a mapping between them.
• In the Boolean algebra case, there is implicit in the definition an intuitively natural partial order "≤" (see Template:W on Wikipedia) such that, given two input tuples a = (a₁, a₂,... a_n) and b = (b₁, b₂,... b_n), a ≤ b means that b can be obtained from a via a series of (zero or more) steps each switching an input from 0 to 1. With this partial order in mind, (only) property (1) of the order theory definition applies.

Synonyms

• Template:Sense Template:L function

Hypernyms

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Hyponyms

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  • Template:Sense
    • Template:L, Template:L, Template:L, Template:L nondecreasing function
    • Template:L, Template:L, Template:L, Template:L increasing function
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    • Template:L, Template:L, Template:L, Template:L nonincreasing function
    • Template:L, Template:L, Template:L, Template:L decreasing function
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  • Template:L, Template:L
  • Template:L, Template:L, Template:L

Related terms

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Translations

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• German: Template:T
• Serbo-Croatian: Template:T
• Turkish: Template:T

See also

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Further reading

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• Monotone function on Template:W
• Monotonic Function on Template:W