In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
Monotonicity in calculus and analysis
In calculus, a function $f$ defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing, increasing or non-decreasing), if for all $x$ and $y$ such that $x\; \backslash leq\; y$ one has $f\backslash !\backslash left(x\backslash right)\; \backslash leq\; f\backslash !\backslash left(y\backslash right)$, so $f$ preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing, or non-increasing) if, whenever $x\; \backslash leq\; y$, then $f\backslash !\backslash left(x\backslash right)\; \backslash geq\; f\backslash !\backslash left(y\backslash right)$, so it reverses the order (see Figure 2).
If the order $\backslash leq$ in the definition of monotonicity is replaced by the strict order $<$, then one obtains a stronger requirement. A function with this property is called strictly increasing. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. Functions that are strictly increasing or decreasing are one-to-one (because for $x$ not equal to $y$, either $x\; <\; y$ or $x\; >\; y$ and so, by monotonicity, either $f\backslash !\backslash left(x\backslash right)\; <\; f\backslash !\backslash left(y\backslash right)$ or $f\backslash !\backslash left(x\backslash right)\; >\; f\backslash !\backslash left(y\backslash right)$, thus $f\backslash !\backslash left(x\backslash right)$ is not equal to $f\backslash !\backslash left(y\backslash right)$.)
When functions between discrete sets are considered in combinatorics, it is not always obvious that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, so one finds the terms weakly increasing and weakly decreasing to stress this possibility.
The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the function of figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing.
The term monotonic transformation can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. Notably, this is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences).^{[1]}
A function $f\backslash !\backslash left(x\backslash right)$ is said to be absolutely monotonic over an interval $\backslash left(a,\; b\backslash right)$ if the derivatives of all orders of $f$ are nonnegative at all points on the interval.
Some basic applications and results
The following properties are true for a monotonic function $f\backslash colon\; \backslash mathbb\{R\}\; \backslash to\; \backslash mathbb\{R\}$:
- $f$ has limits from the right and from the left at every point of its domain;
- $f$ has a limit at positive or negative infinity ( $\backslash pm\backslash infty$ ) of either a real number, $\backslash infty$, or $\backslash left(-\backslash infty\backslash right)$.
- $f$ can only have jump discontinuities;
- $f$ can only have countably many discontinuities in its domain.
These properties are the reason why monotonic functions are useful in technical work in analysis. Two facts about these functions are:
- if $f$ is a monotonic function defined on an interval $I$, then $f$ is differentiable almost everywhere on $I$, i.e. the set $\backslash left\backslash \{x:\; x\; \backslash in\; I\backslash right\backslash \}$ of numbers $x$ in $I$ such that $f$ is not differentiable in $x$ has Lebesgue measure zero.
- if $f$ is a monotonic function defined on an interval $\backslash left[a,\; b\backslash right]$, then $f$ is Riemann integrable.
An important application of monotonic functions is in probability theory. If $X$ is a random variable, its cumulative distribution function $F\_X\backslash !\backslash left(x\backslash right)\; =\; \backslash text\{Prob\}\backslash !\backslash left(X\; \backslash leq\; x\backslash right)$ is a monotonically increasing function.
A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing.
When $f$ is a strictly monotonic function, then $f$ is injective on its domain, and if $T$ is the range of $f$, then there is an inverse function on $T$ for $f$.
Monotonicity in functional analysis
In functional analysis on a topological vector space X, a (possibly non-linear) operator T : X → X^{∗} is said to be a monotone operator if
- $(Tu\; -\; Tv,\; u\; -\; v)\; \backslash geq\; 0\; \backslash quad\; \backslash forall\; u,v\; \backslash in\; X.$
Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.
A subset G of X × X^{∗} is said to be a monotone set if for every pair [u_{1},w_{1}] and [u_{2},w_{2}] in G,
- $(w\_1\; -\; w\_2,\; u\_1\; -\; u\_2)\; \backslash geq\; 0.$
G is said to be maximal monotone if it is maximal among all monotone sets in the sense of set inclusion. The graph of a monotone operator G(T) is a monotone set. A monotone operator is said to be maximal monotone if its graph is a maximal monotone set.
Monotonicity in order theory
Order theory deals with arbitrary partially ordered sets and preordered sets in addition to real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not total. Furthermore, the strict relations < and > are of little use in many non-total orders and hence no additional terminology is introduced for them.
A monotone function is also called isotone, or order-preserving. The dual notion is often called antitone, anti-monotone, or order-reversing. Hence, an antitone function f satisfies the property
- x ≤ y implies f(x) ≥ f(y),
for all x and y in its domain. It is easy to see that the composite of two monotone mappings is also monotone.
A constant function is both monotone and antitone; conversely, if f is both monotone and antitone, and if the domain of f is a lattice, then f must be constant.
Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are order embeddings (functions for which x ≤ y if and only if f(x) ≤ f(y)) and order isomorphisms (surjective order embeddings).
Monotonicity in the context of search algorithms
In the context of search algorithms monotonicity (also called consistency) is a condition applied to heuristic functions. A heuristic h(n) is monotonic if, for every node n and every successor n' of n generated by any action a, the estimated cost of reaching the goal from n is no greater than the step cost of getting to n' plus the estimated cost of reaching the goal from n' ,
- $h(n)\; \backslash leq\; c(n,\; a,\; n\text{'})\; +\; h(n\text{'}).$
This is a form of triangle inequality, with n, n', and the goal G_{n} closest to n. Because every monotonic heuristic is also admissible, monotonicity is a stricter requirement than admissibility. In some heuristic algorithms, such as A*, the algorithm can be considered optimal if it is monotonic.^{[2]}
Boolean functions
In Boolean algebra, a monotonic function is one such that for all a_{i} and b_{i} in {0,1}, if a_{1} ≤ b_{1}, a_{2} ≤ b_{2}, ..., a_{n} ≤ b_{n}, then f(a_{1}, ..., a_{n}) ≤ f(b_{1}, ..., b_{n}). In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that a Boolean function is monotonic when in its Hasse diagram (dual of its Venn diagram), there is no 1 (red vertex) connected to a higher 0 (white vertex).
The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators and and or (in particular not is forbidden). For instance "at least two of a,b,c hold" is a monotonic function of a,b,c, since it can be written for instance as ((a and b) or (a and c) or (b and c)).
The number of such functions on n variables is known as the Dedekind number of n.
Monotonic logic
Monotonicity of entailment is a property of many logic systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. Any true statement in a logic with this property continues to be true, even after adding new axioms. Logics with this property may be called monotonic, to differentiate them from non-monotonic logic.
See also
Notes
Bibliography
External links
- Template:Springer
- Wolfram Demonstrations Project.
- MathWorld.
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