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100-466: As defined in set theory , the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets , such as the set of real numbers , have no minimum or maximum. In statistics , the corresponding concept is the sample maximum and minimum . A real-valued function f defined on a domain X has a global (or absolute ) maximum point at x , if f ( x ) ≥ f ( x ) for all x in X . Similarly,

200-555: A + b x , {\displaystyle a+bx,} or more precisely, f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) . {\displaystyle f(x_{0})+f'(x_{0})(x-x_{0}).} Thus, from the perspective that "if f is differentiable and has non-vanishing derivative at x 0 , {\displaystyle x_{0},} then it does not attain an extremum at x 0 , {\displaystyle x_{0},} "

300-585: A , b ) {\displaystyle (x_{0}-\delta ,x_{0}+\delta )\subset (a,b)} and such that we have f ( x 0 ) ≥ f ( x ) {\displaystyle f(x_{0})\geq f(x)} for all x {\displaystyle x} with | x − x 0 | < δ {\displaystyle \displaystyle |x-x_{0}|<\delta } . Hence for any h ∈ ( 0 , δ ) {\displaystyle h\in (0,\delta )} we have Since

400-527: A greatest element m , then m is a maximal element of the set, also denoted as max ( S ) {\displaystyle \max(S)} . Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with (respect to order induced by T ), then m is a least upper bound of S in T . Similar results hold for least element , minimal element and greatest lower bound . The maximum and minimum function for sets are used in databases , and can be computed rapidly, since

500-434: A mathematical function as a relation from one set (the domain ) to another set (the range ). Fermat%27s theorem (stationary points) In real analysis , Fermat's theorem (also known as interior extremum theorem ) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative

600-399: A necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative , if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum. One way to state Fermat's theorem is that, if a function has a local extremum at some point and is differentiable there, then

700-401: A bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction ). In two and more dimensions, this argument fails. This is illustrated by the function whose only critical point is at (0,0), which is

800-616: A cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness, Thomas Forster has argued that it does reflect an iterative conception of set . Systems of constructive set theory , such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic . Yet other systems accept classical logic but feature

900-424: A domain must occur at critical points (or points where the derivative equals zero). However, not all critical points are extrema. One can often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the first derivative test , second derivative test , or higher-order derivative test , given sufficient differentiability. For any function that is defined piecewise , one finds

1000-561: A foundation for modern mathematics, there has been support for the idea of introducing the basics of naive set theory early in mathematics education . In the US in the 1960s, the New Math experiment aimed to teach basic set theory, among other abstract concepts, to primary school students, but was met with much criticism. The math syllabus in European schools followed this trend, and currently includes

1100-495: A foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity , and has various applications in computer science (such as in the theory of relational algebra ), philosophy , formal semantics , and evolutionary dynamics . Its foundational appeal, together with its paradoxes , and its implications for

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1200-422: A larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models . A cardinal invariant

1300-477: A local maximum then, roughly, there is a (possibly small) neighborhood of x 0 {\displaystyle \displaystyle x_{0}} such as the function "is increasing before" and "decreasing after" x 0 {\displaystyle \displaystyle x_{0}} . As the derivative is positive for an increasing function and negative for a decreasing function, f ′ {\displaystyle \displaystyle f'}

1400-418: A local minimum with f (0,0) = 0. However, it cannot be a global one, because f (2,3) = −5. If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a functional ), then the extremum is found using the calculus of variations . Maxima and minima can also be defined for sets. In general, if an ordered set S has

1500-442: A maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is greatest (or least). For a practical example, assume a situation where someone has 200 {\displaystyle 200} feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where x {\displaystyle x} is the length, y {\displaystyle y}

1600-551: A maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum. If an infinite chain S is bounded, then the closure Cl ( S ) of the set occasionally has a minimum and a maximum, in which case they are called the greatest lower bound and the least upper bound of the set S , respectively. Set theory Set theory is the branch of mathematical logic that studies sets , which can be informally described as collections of objects. Although objects of any kind can be collected into

1700-407: A member and a proper subset of the set {1, {1}} . Just as arithmetic features binary operations on numbers , set theory features binary operations on sets. The following is a partial list of them: Some basic sets of central importance are the set of natural numbers , the set of real numbers and the empty set —the unique set containing no elements. The empty set is also occasionally called

1800-428: A minimum point. An important example is a function whose domain is a closed and bounded interval of real numbers (see the graph above). Finding global maxima and minima is the goal of mathematical optimization . If a function is continuous on a closed interval, then by the extreme value theorem , global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in

1900-485: A model V of ZF satisfies the continuum hypothesis or the axiom of choice , the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent. The study of inner models is common in the study of determinacy and large cardinals , especially when considering axioms such as

2000-476: A natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone spaces . An active area of research is the univalent foundations and related to it homotopy type theory . Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types . Principles such as

2100-712: A nonstandard membership relation. These include rough set theory and fuzzy set theory , in which the value of an atomic formula embodying the membership relation is not simply True or False . The Boolean-valued models of ZFC are a related subject. An enrichment of ZFC called internal set theory was proposed by Edward Nelson in 1977. Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs , manifolds , rings , vector spaces , and relational algebras can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and

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2200-428: A pure set X {\displaystyle X} is defined to be the least ordinal that is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α {\displaystyle \alpha } , the set V α {\displaystyle V_{\alpha }}

2300-432: A set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms minimum and maximum . If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have

2400-454: A set, set theory — as a branch of mathematics — is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under

2500-834: A spectacular blunder in Remarks on the Foundations of Mathematics : Wittgenstein attempted to refute Gödel's incompleteness theorems after having only read the abstract. As reviewers Kreisel , Bernays , Dummett , and Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as Crispin Wright begun to rehabilitate Wittgenstein's arguments. Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism , finite set theory, and computable set theory. Topoi also give

2600-427: Is x 0 {\displaystyle \displaystyle x_{0}} . The theorem (and its proof below) is more general than the intuition in that it does not require the function to be differentiable over a neighbourhood around x 0 {\displaystyle \displaystyle x_{0}} . It is sufficient for the function to be differentiable only in the extreme point. Suppose that f

2700-413: Is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x with x ≠ x , we have f ( x ) > f ( x ) . Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points. A continuous real-valued function with a compact domain always has a maximum point and

2800-462: Is a topological space , since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows: The definition of local minimum point can also proceed similarly. In both the global and local cases, the concept of a strict extremum can be defined. For example, x is a strict global maximum point if for all x in X with x ≠ x , we have f ( x ) > f ( x ) , and x

2900-530: Is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals , measurable cardinals , and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo–Fraenkel set theory . Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from

3000-416: Is a global extremum of f , then one of the following is true: In higher dimensions, exactly the same statement holds; however, the proof is slightly more complicated. The complication is that in 1 dimension, one can either move left or right from a point, while in higher dimensions, one can move in many directions. Thus, if the derivative does not vanish, one must argue that there is some direction in which

3100-408: Is a neighborhood of x 0 {\displaystyle x_{0}} on which the secant lines through x 0 {\displaystyle x_{0}} all have positive slope, and thus to the right of x 0 , {\displaystyle x_{0},} f is greater, and to the left of x 0 , {\displaystyle x_{0},} f

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3200-741: Is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory. Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem

3300-546: Is between set theory and recursion theory . It includes the study of lightface pointclasses , and is closely related to hyperarithmetical theory . In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable. A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations . This has important applications to

3400-464: Is changing. It can only attain a maximum or minimum if it "stops" – if the derivative vanishes (or if it is not differentiable, or if one runs into the boundary and cannot continue). However, making "behaves like a linear function" precise requires careful analytic proof. More precisely, the intuition can be stated as: if the derivative is positive, there is some point to the right of x 0 {\displaystyle x_{0}} where f

3500-414: Is defined to consist of all pure sets with rank less than α {\displaystyle \alpha } . The entire von Neumann universe is denoted  V {\displaystyle V} . Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams . The intuitive approach tacitly assumes that a set may be formed from

3600-405: Is differentiable at x 0 ∈ ( a , b ) , {\displaystyle x_{0}\in (a,b),} with derivative K, and assume without loss of generality that K > 0 , {\displaystyle K>0,} so the tangent line at x 0 {\displaystyle x_{0}} has positive slope (is increasing). Then there

3700-399: Is greater, and some point to the left of x 0 {\displaystyle x_{0}} where f is less, and thus f attains neither a maximum nor a minimum at x 0 . {\displaystyle x_{0}.} Conversely, if the derivative is negative, there is a point to the right which is lesser, and a point to the left which is greater. Stated this way,

3800-916: Is increasing on a neighborhood of x 0 , {\displaystyle x_{0},} as follows. If f ′ ( x 0 ) = K > 0 {\displaystyle f'(x_{0})=K>0} and f ∈ C 1 , {\displaystyle f\in C^{1},} then by continuity of the derivative, there is some ε 0 > 0 {\displaystyle \varepsilon _{0}>0} such that f ′ ( x ) > K / 2 {\displaystyle f'(x)>K/2} for all x ∈ ( x 0 − ε 0 , x 0 + ε 0 ) {\displaystyle x\in (x_{0}-\varepsilon _{0},x_{0}+\varepsilon _{0})} . Then f

3900-664: Is increasing on this interval, by the mean value theorem : the slope of any secant line is at least K / 2 , {\displaystyle K/2,} as it equals the slope of some tangent line. However, in the general statement of Fermat's theorem, where one is only given that the derivative at x 0 {\displaystyle x_{0}} is positive, one can only conclude that secant lines through x 0 {\displaystyle x_{0}} will have positive slope, for secant lines between x 0 {\displaystyle x_{0}} and near enough points. Conversely, if

4000-402: Is it decreasing down to or increasing up from 0 – it oscillates wildly near 0. This pathology can be understood because, while the function g is everywhere differentiable, it is not continuously differentiable: the limit of g ′ ( x ) {\displaystyle g'(x)} as x → 0 {\displaystyle x\to 0} does not exist, so

4100-414: Is lesser. The schematic of the proof is: Formally, by the definition of derivative, f ′ ( x 0 ) = K {\displaystyle f'(x_{0})=K} means that In particular, for sufficiently small ε {\displaystyle \varepsilon } (less than some ε 0 {\displaystyle \varepsilon _{0}} ),

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4200-415: Is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy , based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion ) an ordinal number α {\displaystyle \alpha } , known as its rank. The rank of

4300-476: Is more flexible than a simple yes or no answer and can be a real number such as 0.75. An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether

4400-543: Is often held in the context of Fermat's theorem is to assume that it makes a stronger statement about local behavior than it does. Notably, Fermat's theorem does not say that functions (monotonically) "increase up to" or "decrease down from" a local maximum. This is very similar to the misconception that a limit means "monotonically getting closer to a point". For "well-behaved functions" (which here means continuously differentiable ), some intuitions hold, but in general functions may be ill-behaved, as illustrated below. The moral

4500-492: Is positive before and negative after x 0 {\displaystyle \displaystyle x_{0}} . f ′ {\displaystyle \displaystyle f'} does not skip values (by Darboux's theorem ), so it has to be zero at some point between the positive and negative values. The only point in the neighbourhood where it is possible to have f ′ ( x ) = 0 {\displaystyle \displaystyle f'(x)=0}

4600-604: Is restricted. Since width is positive, then x > 0 {\displaystyle x>0} , and since x = 100 − y {\displaystyle x=100-y} , that implies that x < 100 {\displaystyle x<100} . Plug in critical point 50 {\displaystyle 50} , as well as endpoints 0 {\displaystyle 0} and 100 {\displaystyle 100} , into x y = x ( 100 − x ) {\displaystyle xy=x(100-x)} , and

4700-676: Is that defining sets using the axiom schemas of specification and replacement , as well as the axiom of power set , introduces impredicativity , a type of circularity , into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]". Ludwig Wittgenstein condemned set theory philosophically for its connotations of mathematical platonism . He wrote that "set theory

4800-493: Is that derivatives determine infinitesimal behavior, and that continuous derivatives determine local behavior. If f is continuously differentiable ( C 1 ) {\displaystyle \left(C^{1}\right)} on an open neighborhood of the point x 0 {\displaystyle x_{0}} , then f ′ ( x 0 ) > 0 {\displaystyle f'(x_{0})>0} does mean that f

4900-418: Is the normal Moore space question , a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC. From set theory's inception, some mathematicians have objected to it as a foundation for mathematics . The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from

5000-572: Is the first non-vanishing derivative, and f ( k ) {\displaystyle f^{(k)}} is continuous, so f ∈ C k {\displaystyle f\in C^{k}} ), then one can treat f as locally close to a polynomial of degree k, since it behaves approximately as f ( k ) ( x 0 ) ( x − x 0 ) k , {\displaystyle f^{(k)}(x_{0})(x-x_{0})^{k},} but if

5100-403: Is the subset relation, also called set inclusion . If all the members of set A are also members of set B , then A is a subset of B , denoted A ⊆ B . For example, {1, 2} is a subset of {1, 2, 3} , and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected,

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5200-444: Is the width, and x y {\displaystyle xy} is the area: The derivative with respect to x {\displaystyle x} is: Setting this equal to 0 {\displaystyle 0} reveals that x = 50 {\displaystyle x=50} is our only critical point . Now retrieve the endpoints by determining the interval to which x {\displaystyle x}

5300-702: Is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein identified mathematics with algorithmic human deduction; the need for a secure foundation for mathematics seemed, to him, nonsensical. Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical constructivism and finitism . Meta-mathematical statements — which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory — are not mathematics. Few modern philosophers have adopted Wittgenstein's views after

5400-474: Is zero at that point). Fermat's theorem is named after a French mathematician Pierre de Fermat . By using Fermat's theorem, the potential extrema of a function f {\displaystyle \displaystyle f} , with derivative f ′ {\displaystyle \displaystyle f'} , are found by solving an equation in f ′ {\displaystyle \displaystyle f'} . Fermat's theorem gives only

5500-450: The Axiom of Choice have recently seen applications in evolutionary dynamics , enhancing the understanding of well-established models of evolution and interaction. Set theory is a major area of research in mathematics with many interrelated subfields: Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and

5600-496: The axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results. As set theory gained popularity as

5700-506: The constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop 's influential book Foundations of Constructive Analysis . A different objection put forth by Henri Poincaré

5800-405: The exterior derivative d f {\displaystyle df} is zero. Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the zeros of the derivative), the non-differentiable points, and the boundary points, and then investigating this set to determine

5900-410: The k -th derivative is not continuous, one cannot draw such conclusions, and it may behave rather differently. The function sin ⁡ ( 1 / x ) {\displaystyle \sin(1/x)} oscillates increasingly rapidly between − 1 {\displaystyle -1} and 1 {\displaystyle 1} as x approaches 0. Consequently,

6000-401: The limit of this ratio as h {\displaystyle \displaystyle h} gets close to 0 from above exists and is equal to f ′ ( x 0 ) {\displaystyle \displaystyle f'(x_{0})} we conclude that f ′ ( x 0 ) ≤ 0 {\displaystyle f'(x_{0})\leq 0} . On

6100-404: The natural and real numbers can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms. Set theory as a foundation for mathematical analysis , topology , abstract algebra , and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from

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6200-466: The null set , though this name is ambiguous and can lead to several interpretations. The Power set of a set A , denoted P ( A ) {\displaystyle {\mathcal {P}}(A)} , is the set whose members are all of the possible subsets of A . For example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} } . Notably, P ( A ) {\displaystyle {\mathcal {P}}(A)} contains both A and

6300-412: The (enlargeable) figure on the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of

6400-402: The axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice). A large cardinal

6500-832: The class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox . Axiomatic set theory was originally devised to rid set theory of such paradoxes. The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy . Such systems come in two flavors, those whose ontology consists of: The above systems can be modified to allow urelements , objects that can be members of sets but that are not themselves sets and do not have any members. The New Foundations systems of NFU (allowing urelements ) and NF (lacking them), associate with Willard Van Orman Quine , are not based on

6600-445: The concept of infinity and its multiple applications have made set theory an area of major interest for logicians and philosophers of mathematics . Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals . Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however,

6700-492: The derivative is 0. Suppose that x 0 {\displaystyle \displaystyle x_{0}} is a local maximum (a similar proof applies if x 0 {\displaystyle \displaystyle x_{0}} is a local minimum). Then there exists δ > 0 {\displaystyle \delta >0} such that ( x 0 − δ , x 0 + δ ) ⊂ (

6800-756: The derivative of f at a point is zero ( x 0 {\displaystyle x_{0}} is a stationary point), one cannot in general conclude anything about the local behavior of f – it may increase to one side and decrease to the other (as in x 3 {\displaystyle x^{3}} ), increase to both sides (as in x 4 {\displaystyle x^{4}} ), decrease to both sides (as in − x 4 {\displaystyle -x^{4}} ), or behave in more complicated ways, such as oscillating (as in x 2 sin ⁡ ( 1 / x ) {\displaystyle x^{2}\sin(1/x)} , as discussed below). One can analyze

6900-402: The domain X is a metric space , then f is said to have a local (or relative ) maximum point at the point x , if there exists some ε > 0 such that f ( x ) ≥ f ( x ) for all x in X within distance ε of x . Similarly, the function has a local minimum point at x , if f ( x ) ≤ f ( x ) for all x in X within distance ε of x . A similar definition can be used when X

7000-457: The empty set. A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality

7100-856: The extended function is continuous and everywhere differentiable (it is differentiable at 0 with derivative 0), but has rather unexpected behavior near 0: in any neighborhood of 0 it attains 0 infinitely many times, but also equals 2 x 2 {\displaystyle 2x^{2}} (a positive number) infinitely often. Continuing in this vein, one may define g ( x ) = ( 2 + sin ⁡ ( 1 / x ) ) x 2 {\displaystyle g(x)=(2+\sin(1/x))x^{2}} , which oscillates between x 2 {\displaystyle x^{2}} and 3 x 2 {\displaystyle 3x^{2}} . The function has its local and global minimum at x = 0 {\displaystyle x=0} , but on no neighborhood of 0

7200-398: The extrema. One can do this either by evaluating the function at each point and taking the maximum, or by analyzing the derivatives further, using the first derivative test , the second derivative test , or the higher-order derivative test . Intuitively, a differentiable function is approximated by its derivative – a differentiable function behaves infinitesimally like a linear function

7300-445: The function f ( x ) = ( 1 + sin ⁡ ( 1 / x ) ) x 2 {\displaystyle f(x)=(1+\sin(1/x))x^{2}} oscillates increasingly rapidly between 0 and 2 x 2 {\displaystyle 2x^{2}} as x approaches 0. If one extends this function by defining f ( 0 ) = 0 {\displaystyle f(0)=0} then

7400-632: The function has a global (or absolute ) minimum point at x , if f ( x ) ≤ f ( x ) for all x in X . The value of the function at a maximum point is called the maximum value of the function, denoted max ( f ( x ) ) {\displaystyle \max(f(x))} , and the value of the function at a minimum point is called the minimum value of the function, (denoted min ( f ( x ) ) {\displaystyle \min(f(x))} for clarity). Symbolically, this can be written as follows: The definition of global minimum point also proceeds similarly. If

7500-522: The function increases – and thus in the opposite direction the function decreases. This is the only change to the proof or the analysis. The statement can also be extended to differentiable manifolds . If f : M → R {\displaystyle f:M\to \mathbb {R} } is a differentiable function on a manifold M {\displaystyle M} , then its local extrema must be critical points of f {\displaystyle f} , in particular points where

7600-477: The function's derivative at that point must be zero. In precise mathematical language: Another way to understand the theorem is via the contrapositive statement: if the derivative of a function at any point is not zero, then there is not a local extremum at that point. Formally: The global extrema of a function f on a domain A occur only at boundaries , non-differentiable points, and stationary points. If x 0 {\displaystyle x_{0}}

7700-446: The infinitesimal behavior via the second derivative test and higher-order derivative test , if the function is differentiable enough, and if the first non-vanishing derivative at x 0 {\displaystyle x_{0}} is a continuous function , one can then conclude local behavior (i.e., if f ( k ) ( x 0 ) ≠ 0 {\displaystyle f^{(k)}(x_{0})\neq 0}

7800-400: The interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the greatest (or least) one.Minima For differentiable functions , Fermat's theorem states that local extrema in the interior of

7900-420: The interval to the left, f is less than f ( x 0 ) . {\displaystyle f(x_{0}).} Thus x 0 {\displaystyle x_{0}} is not a local or global maximum or minimum of f. Alternatively, one can start by assuming that x 0 {\displaystyle \displaystyle x_{0}} is a local maximum, and then prove that

8000-437: The intuition is that if the derivative at x 0 {\displaystyle x_{0}} is positive, the function is increasing near x 0 , {\displaystyle x_{0},} while if the derivative is negative, the function is decreasing near x 0 . {\displaystyle x_{0}.} In both cases, it cannot attain a maximum or minimum, because its value

8100-690: The letter A "), which may be useful when learning computer programming , since Boolean logic is used in various programming languages . Likewise, sets and other collection-like objects, such as multisets and lists , are common datatypes in computer science and programming . In addition to that, sets are commonly referred to in mathematical teaching when talking about different types of numbers (the sets N {\displaystyle \mathbb {N} } of natural numbers , Z {\displaystyle \mathbb {Z} } of integers , R {\displaystyle \mathbb {R} } of real numbers , etc.), and when defining

8200-480: The limit (an infinitesimal statement) with an inequality on a neighborhood (a local statement). Thus, rearranging the equation, if ε > 0 , {\displaystyle \varepsilon >0,} then: so on the interval to the right, f is greater than f ( x 0 ) , {\displaystyle f(x_{0}),} and if ε < 0 , {\displaystyle \varepsilon <0,} then: so on

8300-410: The maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self- decomposable aggregation functions . In the case of a general partial order , the least element (i.e., one that is less than all others) should not be confused with a minimal element (nothing is lesser). Likewise, a greatest element of a partially ordered set (poset) is an upper bound of

8400-406: The name of naive set theory . After the discovery of paradoxes within naive set theory (such as Russell's paradox , Cantor's paradox and the Burali-Forti paradox ), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice ) is still the best-known and most studied. Set theory is commonly employed as

8500-734: The other hand, for h ∈ ( − δ , 0 ) {\displaystyle h\in (-\delta ,0)} we notice that but again the limit as h {\displaystyle \displaystyle h} gets close to 0 from below exists and is equal to f ′ ( x 0 ) {\displaystyle \displaystyle f'(x_{0})} so we also have f ′ ( x 0 ) ≥ 0 {\displaystyle f'(x_{0})\geq 0} . Hence we conclude that f ′ ( x 0 ) = 0. {\displaystyle \displaystyle f'(x_{0})=0.} A subtle misconception that

8600-433: The possibility of a saddle point . For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum. In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if

8700-399: The proof is just translating this into equations and verifying "how much greater or less". The intuition is based on the behavior of polynomial functions . Assume that function f has a maximum at x 0 , the reasoning being similar for a function minimum. If x 0 ∈ ( a , b ) {\displaystyle \displaystyle x_{0}\in (a,b)} is

8800-402: The quotient must be at least K / 2 , {\displaystyle K/2,} by the definition of limit. Thus on the interval ( x 0 − ε 0 , x 0 + ε 0 ) {\displaystyle (x_{0}-\varepsilon _{0},x_{0}+\varepsilon _{0})} one has: one has replaced the equality in

8900-488: The real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure. Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additional sets in order to create

9000-507: The relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath , includes human-written, computer-verified derivations of more than 12,000 theorems starting from ZFC set theory, first-order logic and propositional logic . ZFC and

9100-461: The results are 2500 , 0 , {\displaystyle 2500,0,} and 0 {\displaystyle 0} respectively. Therefore, the greatest area attainable with a rectangle of 200 {\displaystyle 200} feet of fencing is 50 × 50 = 2500 {\displaystyle 50\times 50=2500} . For functions of more than one variable, similar conditions apply. For example, in

9200-477: The set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A ), then m = b . Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable. In a totally ordered set, or chain , all elements are mutually comparable, so such

9300-439: The start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of

9400-401: The study of invariants in many fields of mathematics. In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people"

9500-765: The study of extensions of Ramsey's theorem such as the Erdős–Rado theorem . Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces . It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy . Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals. The field of effective descriptive set theory

9600-478: The subject at different levels in all grades. Venn diagrams are widely employed to explain basic set-theoretic relationships to primary school students (even though John Venn originally devised them as part of a procedure to assess the validity of inferences in term logic ). Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition ) of sets (e.g. "months starting with

9700-403: The term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B , but A is not equal to B . Also, 1, 2, and 3 are members (elements) of the set {1, 2, 3} , but are not subsets of it; and in turn, the subsets, such as {1} , are not members of the set {1, 2, 3} . More complicated relations can exist; for example, the set {1} is both

9800-456: The theory of mathematical relations can be described in set theory. Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica , it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic . For example, properties of

9900-619: Was founded by a single paper in 1874 by Georg Cantor : " On a Property of the Collection of All Real Algebraic Numbers ". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity . Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874, and

10000-532: Was motivated by Cantor's work in real analysis . Set theory begins with a fundamental binary relation between an object o and a set A . If o is a member (or element ) of A , the notation o ∈ A is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets

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