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Intransitivity
In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive. Intransitivity A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C. Some authors call a relation if it is not transitive, that is, (if the relation in question is named R) \lnot\left(\forall a, b, c: a R b \land b R c \implies a R c\right). This statement is equivalent to \exists a,b,c : a R b \land b R c \land \lnot(a R c). For instance, in the food chain, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass. Thus, the relation among life forms is intransitive, in this sense. Another example that does not involve preference loops arises in freemasonry: in some instances lodge A recognizes lodge B, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Relation
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie. On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then this does not imply that Alice is the birth parent of Claire. What is more, it is antitransitive: Alice can ''never'' be the birth parent of Claire. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voting Paradox
The Condorcet paradox (also known as the voting paradox or the paradox of voting) in social choice theory is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic, even if the preferences of individual voters are not cyclic. This is paradoxical, because it means that majority wishes can be in conflict with each other: Suppose majorities prefer, for example, candidate A over B, B over C, and yet C over A. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals. Thus an expectation that transitivity on the part of all individuals' preferences should result in transitivity of societal preferences is an example of a fallacy of composition. The paradox was independently discovered by Lewis Carroll and Edward J. Nanson, but its significance was not recognized until popularized by Duncan Black in the 1940s. Example Suppose we have three candidates, A, B, and C, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a ''directed acyclic graph''. A connected graph without cycles is called a '' tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes what's viewed as basic elements in the economy, including individual agents and markets, their interactions, and the outcomes of interactions. Individual agents may include, for example, households, firms, buyers, and sellers. Macroeconomics analyzes the economy as a system where production, consumption, saving, and investment interact, and factors affecting it: employment of the resources of labour, capital, and land, currency inflation, economic growth, and public policies that have impact on these elements. Other broad distinctions within economics include those between positive economics, describing "what is", and normative economics, advocating "what ought to be"; between economic theory and applied economics; between rational a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taste (sociology)
In sociology, taste or palate is an individual or a demographic group's subjective preferences of dietary, design, cultural and/or aesthetic patterns. Taste manifests socially via distinctions in consumer choices such as delicacies/ beverages, fashions, music, etiquettes, goods, styles of artwork, and other related cultural activities. The social inquiry of taste is about the arbitrary human ability to judge what is considered beautiful, good, proper and valuable. Social and cultural phenomena concerning taste are closely associated to social relations and dynamics between people. The concept of social taste is therefore rarely separated from its accompanying sociological concepts. An understanding of taste as something that is expressed in actions between people helps to perceive many social phenomena that would otherwise be inconceivable. Aesthetic preferences and attendance to various cultural events are associated with education and social origin. Different so ... [...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 (Lichtenstein & Slovic, 2006). The term is also used to mean evaluative judgment in the sense of lik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Value System
In ethics and social sciences, value denotes the degree of importance of something or action, with the aim of determining which actions are best to do or what way is best to live (normative ethics in ethics), or to describe the significance of different actions. Value systems are prospective and prescriptive beliefs; they affect the ethical behavior of a person or are the basis of their intentional activities. Often primary values are strong and secondary values are suitable for changes. What makes an action valuable may in turn depend on the ethical values of the objects it increases, decreases, or alters. An object with "ethic value" may be termed an "ethic or philosophic good" (noun sense). Values can be defined as broad preferences concerning appropriate courses of actions or outcomes. As such, values reflect a person's sense of right and wrong or what "ought" to be. "Equal rights for all", "Excellence deserves admiration", and "People should be treated with respect and digni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Psychology
Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries between the natural and social sciences. Psychologists seek an understanding of the emergent properties of brains, linking the discipline to neuroscience. As social scientists, psychologists aim to understand the behavior of individuals and groups.Fernald LD (2008)''Psychology: Six perspectives'' (pp.12–15). Thousand Oaks, CA: Sage Publications.Hockenbury & Hockenbury. Psychology. Worth Publishers, 2010. Ψ (''psi''), the first letter of the Greek word ''psyche'' from which the term psychology is derived (see below), is commonly associated with the science. A professional practitioner or researcher involved in the discipline is called a psychologist. Some psychologists can also be classified as behavioral or cognitive scientists. Some psychol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intransitive Dice
A set of dice is intransitive (or nontransitive) if it contains three dice, ''A'', ''B'', and ''C'', with the property that ''A'' rolls higher than ''B'' more than half the time, and ''B'' rolls higher than ''C'' more than half the time, but it is not true that ''A'' rolls higher than ''C'' more than half the time. In other words, a set of dice is intransitive if the binary relation – rolls a higher number than more than half the time – on its elements is not transitive. More simply, ''A'' normally beats ''B'', ''B'' normally beats ''C'', but ''A'' does not normally beat ''C''. It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. This is different in that instead of only "''A'' does not normally beat ''C''" it is now "''C'' normally beats ''A"'' Using such a set of dice, one can invent games which are biased in ways that people unused to intransi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condorcet Voting
A Condorcet method (; ) is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, that is, a candidate preferred by more voters than any others, whenever there is such a candidate. A candidate with this property, the ''pairwise champion'' or ''beats-all winner'', is formally called the ''Condorcet winner''. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking. Some elections may not yield a Condorcet winner because voter preferences may be cyclic—that is, it is possible (but rare) that every candidate has an opponent that defeats them in a two-candidate contest.(This is similar to the game rock paper scissors, where each hand shape wins against one opponent and loses to another one). The possibility of such cyclic preferences is known as the Condorcet paradox. However, a smallest group of candidates that beat al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
