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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] |
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 ... [...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 intransiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Jargon
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense. Some phrases, like "in general", appear below in more than one section. Philosophy of mathematics ; abstract nonsense:A tongue-in-cheek reference to category theory, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it's also known as ''general abstract nonsense'' or ''generalized abstract nonsense''. ; canonical:A reference to a standard or choice-free presentation of some mathematical object (e.g., canonical map, canonica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preference (economics)
In economics and other social sciences, preference is the order that an agent gives to alternatives based on their relative utility. A process which results in an "optimal choice" (whether real or theoretical). Preferences are evaluations and concern matters of value, typically in relation to practical reasoning. The character of the preferences is determined purely by a person's tastes instead of the good's prices, personal income, and the availability of goods. However, people are still expected to act in their best (rational) interest. Rationality, in this context, means that when individuals are faced with a choice, they would select the option that maximizes self-interest. Moreover, in every set of alternatives, preferences arise. The belief of preference plays a key role in many disciplines, including moral philosophy and decision theory. The logical properties that preferences possess have major effects also on rational choice theory, which has a carryover effect on a ... [...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 intransiti ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ... [...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 ratio ... [...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 socioecono ... [...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 liking ... [...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 d ... [...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 ps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
