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Contour Set
In mathematics, contour sets generalize and formalize the everyday notions of *everything superior to something *everything superior or equivalent to something *everything inferior to something *everything inferior or equivalent to something. Formal definitions Given a relation on pairs of elements of set X :\succcurlyeq~\subseteq~X^2 and an element x of X :x\in X The upper contour set of x is the set of all y that are related to x: :\left\ The lower contour set of x is the set of all y such that x is related to them: :\left\ The strict upper contour set of x is the set of all y that are related to x without x being ''in this way'' related to any of them: :\left\ The strict lower contour set of x is the set of all y such that x is related to them without any of them being ''in this way'' related to x: :\left\ The formal expressions of the last two may be simplified if we have defined :\succ~=~\left\ so that a is related to b but b is ''not'' related to a, in which case the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interactions of Agent (economics), economic agents and how economy, economies work. Microeconomics analyses what is viewed as basic elements within economy, economies, including individual agents and market (economics), markets, their interactions, and the outcomes of interactions. Individual agents may include, for example, households, firms, buyers, and sellers. Macroeconomics analyses economies as systems where production, distribution, consumption, savings, and Expenditure, investment expenditure interact; and the factors of production affecting them, such as: Labour (human activity), labour, Capital (economics), capital, Land (economics), land, and Entrepreneurship, enterprise, inflation, economic growth, and public policies that impact gloss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypograph (mathematics)
In mathematics, the hypograph or subgraph of a function f:\R^\rightarrow \R is the set of points lying on or below its graph. A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph. The domain (rather than the codomain) of the function is not particularly important for this definition; it can be an arbitrary set instead of \mathbb^n. Definition The definition of the hypograph was inspired by that of the graph of a function, where the of f : X \to Y is defined to be the set :\operatorname f := \left\. The or of a function f : X \to \infty, \infty/math> valued in the extended real numbers \infty, \infty= \mathbb \cup \ is the set : \begin \operatorname f &= \left\ \\ &= \left f^(\infty) \times \mathbb \right\cup \bigcup_ (\ \times (-\infty, f(x)]). \end Similarly, the set of points on or above the function is its epigraph. The is the hypograph with the graph removed: : \begin \operatorname_S f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epigraph (mathematics)
In mathematics, the epigraph or supergraph of a Function (mathematics), function f : X \to [-\infty, \infty] valued in the Extended real number line, extended real numbers [-\infty, \infty] = \Reals \cup \ is the Set (mathematics), set \operatorname f = \ consisting of all points in the Cartesian product X \times \Reals lying on or above the function's Graph of a function, graph. Similarly, the strict epigraph \operatorname_S f is the set of points in X \times \Reals lying strictly above its graph. Importantly, unlike the graph of f, the epigraph consists of points in X \times \Reals (this is true of the graph only when f is real-valued). If the function takes \pm \infty as a value then \operatorname f will be a subset of its epigraph \operatorname f. For example, if f\left(x_0\right) = \infty then the point \left(x_0, f\left(x_0\right)\right) = \left(x_0, \infty\right) will belong to \operatorname f but not to \operatorname f. These two sets are nevertheless closely re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a Set (mathematics), set , often denoted by A^c (or ), is the set of Element (mathematics), elements not in . When all elements in the Universe (set theory), universe, i.e. all elements under consideration, are considered to be Element (mathematics), members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a normative context, utility refers to a goal or objective that we wish to maximize, i.e., an objective function. This kind of utility bears a closer resemblance to the original utilitarian concept, developed by moral philosophers such as Jeremy Bentham and John Stuart Mill. * In a descriptive context, the term refers to an ''apparent'' objective function; such a function is revealed by a person's behavior, and specifically by their preferences over lotteries, which can be any quantified choice. The relationship between these two kinds of utility functions has been a source of controversy among both economists and ethicists, with most maintaining that the two are distinct but generally related. Utility function Consider a set of alternatives among which a person has a preference ordering. A utility fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preference
In psychology, economics and philosophy, preference is a technical term usually used in relation to choosing between alternatives. For example, someone prefers A over B if they would rather choose A than B. Preferences are central to decision theory because of this relation to behavior. Some methods such as Ordinal Priority Approach use preference relation for decision-making. As connative states, they are closely related to desires. The difference between the two is that desires are directed at one object while preferences concern a comparison between two alternatives, of which one is preferred to the other. In insolvency, the term is used to determine which outstanding obligation the insolvent party has to settle first. Psychology In psychology, preferences refer to an individual's attitude towards a set of objects, typically reflected in an explicit decision-making process. The term is also used to mean evaluative judgment in the sense of liking or disliking an object, as in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outcome (game Theory)
In game theory, the outcome of a game is the ultimate result of a strategic interaction with one or more people, dependant on the choices made by all participants in a certain exchange. It represents the final payoff resulting from a set of actions that individuals can take within the context of the game. Outcomes are pivotal in determining the payoffs and expected utility for parties involved. Game theorists commonly study how the outcome of a game is determined and what factors affect it. A strategy is a set of actions that a player can take in response to the actions of others. Each player’s strategy is based on their expectation of what the other players are likely to do, often explained in terms of probability. Outcomes are dependent on the combination of strategies chosen by involved players and can be represented in a number of ways; one common way is a payoff matrix showing the individual payoffs for each players with a combination of strategies, as seen in the payoff mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Good (economics And Accounting)
In economics, goods are anything that is good, usually in the sense that it provides well-being, welfare or utility to someone.Alan V. Deardorff, 2006. ''Terms Of Trade: Glossary of International Economics'', World Scientific. Online version: Deardorffs' Glossary of International Economics"good" an Goods can be contrasted with bads, i.e. things that provide negative value for users, like chore division, chores or waste. A bad lowers a consumer's overall welfare. Economics focuses on the study of economic goods, i.e. goods that are scarce; in other words, producing the good requires expending effort or resources. Economic goods contrast with free goods such as air, for which there is an unlimited supply.Samuelson, P. Anthony., Samuelson, W. (1980). Economics. 11th ed. / New York: McGraw-Hill. Goods are the result of the Secondary sector of the economy which involves the transformation of raw materials or intermediate goods into goods. Utility and characteristics of goods The c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Notation
Mathematical notation consists of using glossary of mathematical symbols, symbols for representing operation (mathematics), operations, unspecified numbers, relation (mathematics), relations, and any other mathematical objects and assembling them into expression (mathematics), expressions and formulas. Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and property (philosophy), properties in a concise, unambiguous, and accurate way. For example, the physicist Albert Einstein's formula E=mc^2 is the quantitative representation in mathematical notation of mass–energy equivalence. Mathematical notation was first introduced by François Viète at the end of the 16th century and largely expanded during the 17th and 18th centuries by René Descartes, Isaac Newton, Gottfried Wilhelm Leibniz, and overall Leonhard Euler. Symbols and typeface The use of many symbols is the basis of mathematical notation. They play a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
