monotone function

Noun

 * 1)  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).
 * 2)  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)  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.
 * 1)  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).
 * 2)  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.
 * 1)  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).
 * 2)  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 and, 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  on Wikipedia) such that, given two input tuples a = (a1, a2,... an) and b = (b1, b2,... bn), 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

 * function

Hyponyms

 * ,, , nondecreasing function
 * ,, , increasing function
 * ,, , nonincreasing function
 * ,, , decreasing function
 * ,, , nonincreasing function
 * ,, , decreasing function
 * ,, , decreasing function

Translations

 * German: monotone Funktion
 * Serbo-Croatian: monotona funkcija
 * Turkish: monoton fonksiyon


 * German: monotone Funktion
 * Serbo-Croatian: monotona funkcija
 * Turkish: monoton fonksiyon