Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.^[1]^[2]^[3] This concept first arose in calculus, and was later generalized đồ sộ the more abstract setting of order theory.

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In calculus and analysis[edit]

In calculus, a function defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing.^[2] That is, as per Fig. 1, a function that increases monotonically does not exclusively have đồ sộ increase, it simply must not decrease.

A function is called monotonically increasing (also increasing or non-decreasing)^[3] if for all and such that one has , so sánh preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing)^[3] if, whenever , then , so sánh it reverses the order (see Figure 2).

If the order in the definition of monotonicity is replaced by the strict order , one obtains a stronger requirement. A function with this property is called strictly increasing (also increasing).^[3]^[4] Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing (also decreasing).^[3]^[4] A function with either property is called strictly monotone. Functions that are strictly monotone are one-to-one (because for not equal đồ sộ , either or and so sánh, by monotonicity, either or , thus .)

To avoid ambiguity, the terms weakly monotone, weakly increasing and weakly decreasing are often used đồ sộ refer đồ sộ non-strict monotonicity.

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 non-monotonic function shown in 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.

A function is said đồ sộ be absolutely monotonic over an interval if the derivatives of all orders of are nonnegative or all nonpositive at all points on the interval.

Inverse of function[edit]

All strictly monotonic functions are invertible because they are guaranteed đồ sộ have a one-to-one mapping from their range đồ sộ their tên miền.

However, functions that are only weakly monotone are not invertible because they are constant on some interval (and therefore are not one-to-one).

A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it is not strictly monotonic everywhere. For example, if is strictly increasing on the range , then it has an inverse on the range .

Note that the term monotonic is sometimes used in place of strictly monotonic, so sánh a source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible.^{[citation needed]}

Monotonic transformation[edit]

The term monotonic transformation (or monotone transformation) may also cause confusion because it refers đồ sộ a transformation by a strictly increasing function. This is the case in economics with respect đồ sộ the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences).^[5] In this context, the term "monotonic transformation" refers đồ sộ a positive monotonic transformation and is intended đồ sộ distinguish it from a "negative monotonic transformation," which reverses the order of the numbers.^[6]

Some basic applications and results[edit]

The following properties are true for a monotonic function :

These properties are the reason why monotonic functions are useful in technical work in analysis. Other important properties of these functions include:

An important application of monotonic functions is in probability theory. If is a random variable, its cumulative distribution function is a monotonically increasing function.

A function is unimodal if it is monotonically increasing up đồ sộ some point (the mode) and then monotonically decreasing.

When is a strictly monotonic function, then is injective on its tên miền, and if is the range of , then there is an inverse function on for . In contrast, each constant function is monotonic, but not injective,^[7] and hence cannot have an inverse.

The graphic shows six monotonic functions. Their simplest forms are shown in the plot area and the expressions used đồ sộ create them are shown on the y-axis.

In topology[edit]

A map is said đồ sộ be monotone if each of its fibers is connected; that is, for each element the (possibly empty) mix is a connected subspace of

In functional analysis[edit]

In functional analysis on a topological vector space , a (possibly non-linear) operator is said đồ sộ be a monotone operator if

Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.

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A subset of is said đồ sộ be a monotone set if for every pair and in ,

is said đồ sộ be maximal monotone if it is maximal among all monotone sets in the sense of mix inclusion. The graph of a monotone operator is a monotone mix. A monotone operator is said đồ sộ be maximal monotone if its graph is a maximal monotone set.

In order theory[edit]

Order theory giao dịch with arbitrary partially ordered sets and preordered sets as a generalization of 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 đồ sộ 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.

Letting denote the partial order relation of any partially ordered mix, a monotone function, also called isotone, or order-preserving, satisfies the property

for all x and y in its tên miền. The composite of two monotone mappings is also monotone.

The dual notion is often called antitone, anti-monotone, or order-reversing. Hence, an antitone function f satisfies the property

for all x and y in its tên miền.

A constant function is both monotone and antitone; conversely, if f is both monotone and antitone, and if the tên miền 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 if and only if and order isomorphisms (surjective order embeddings).

In the context of tìm kiếm algorithms[edit]

In the context of tìm kiếm algorithms monotonicity (also called consistency) is a condition applied đồ sộ heuristic functions. A heuristic 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 thở the step cost of getting đồ sộ n' plus the estimated cost of reaching the goal from n',

This is a size of triangle inequality, with n, n', and the goal G_n closest đồ sộ n. Because every monotonic heuristic is also admissible, monotonicity is a stricter requirement than thở admissibility. Some heuristic algorithms such as A* can be proven optimal provided that the heuristic they use is monotonic.^[8]

In Boolean functions[edit]

In Boolean algebra, a monotonic function is one such that for all a_i and b_i in {0,1}, if a₁ ≤ b₁, a₂ ≤ b₂, ..., a_n ≤ b_n (i.e. the Cartesian product {0, 1}ⁿ is ordered coordinatewise), then f(a₁, ..., a_n) ≤ f(b₁, ..., b_n). In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false đồ sộ true can only cause the output đồ sộ switch from false đồ sộ true and not from true đồ sộ false. Graphically, this means that an n-ary Boolean function is monotonic when its representation as an n-cube labelled with truth values has no upward edge from true đồ sộ false. (This labelled Hasse diagram is the dual of the function's labelled Venn diagram, which is the more common representation for n ≤ 3.)

The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than thở 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.

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Notes[edit]

^ Clapham, Christopher; Nicholson, James (2014). Oxford Concise Dictionary of Mathematics (5th ed.). Oxford University Press.
^ ^a ^b Stover, Christopher. "Monotonic Function". Wolfram MathWorld. Retrieved 2018-01-29.
^ ^a ^b ^c ^d ^e "Monotone function". Encyclopedia of Mathematics. Retrieved 2018-01-29.
^ ^a ^b Spivak, Michael (1994). Calculus. 1572 West Gray, #377 Houston, Texas 77019: Publish or Perish, Inc. p. 192. ISBN 0-914098-89-6.{{cite book}}: CS1 maint: location (link)
^ See the section on Cardinal Versus Ordinal Utility in Simon & Blume (1994).
^ Varian, Hal R. (2010). Intermediate Microeconomics (8th ed.). W. W. Norton & Company. p. 56. ISBN 9780393934243.
^ if its tên miền has more than thở one element
^ Conditions for optimality: Admissibility and consistency pg. 94–95 (Russell & Norvig 2010).

Bibliography[edit]

Bartle, Robert G. (1976). The elements of real analysis (second ed.).
Grätzer, George (1971). Lattice theory: first concepts and distributive lattices. ISBN 0-7167-0442-0.
Pemberton, Malcolm; Rau, Nicholas (2001). Mathematics for economists: an introductory textbook. Manchester University Press. ISBN 0-7190-3341-1.
Renardy, Michael & Rogers, Robert C. (2004). An introduction đồ sộ partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0.
Riesz, Frigyes & Béla Szőkefalvi-Nagy (1990). Functional Analysis. Courier Dover Publications. ISBN 978-0-486-66289-3.
Russell, Stuart J.; Norvig, Peter (2010). Artificial Intelligence: A Modern Approach (3rd ed.). Upper Saddle River, New Jersey: Prentice Hall. ISBN 978-0-13-604259-4.
Simon, Carl Phường.; Blume, Lawrence (April 1994). Mathematics for Economists (first ed.). ISBN 978-0-393-95733-4. (Definition 9.31)

External links[edit]

"Monotone function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Convergence of a Monotonic Sequence by Anik Debnath and Thomas Roxlo (The Harker School), Wolfram Demonstrations Project.
Weisstein, Eric W. "Monotonic Function". MathWorld.