relation

Some useful properties of inequations in algebra are: Any quantity can be added to both sides. Any quantity can be subtracted from both sides. Any nonzero quantity can be multiplied to both sides. Any nonzero quantity can divide both sides. Generally, any injective function can be applied to both sides. Property (5) is somewhat of a tautology, since injective functions may be defined as fu ...

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Inequation, Inequation - Properties

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An input to a function is called argument of the function. For each argument x, the corresponding unique y in the codomain is called the function value at x, or the image of x under f. The image of x can be written as f(x) or as y. Written mathematics sometimes omits the parentheses around the argument, thus: sin x, but calculators and computers require parentheses around the argument. In some branches of mathematics, such as automata theory, th ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - The vocabulary of functions, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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It is not sufficiently recognized that Peirce’s career was that of a scientist, not a philosopher; and that during his lifetime he was known and valued chiefly as a scientist, only secondly as a logician, and scarcely at all as a philosopher. Even his work in philosophy and logic will not be understood until this fact becomes a standing premise of Peircian studies. (Max Fisch, in Moore and Robin 1964: 486). Upon this first, and in one sense sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not ...

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Charles Peirce, Charles Peirce - Life, Charles Peirce - Reception, Charles Peirce - Works, Charles Peirce - Major publications, Charles Peirce - Peirce's philosophy, Charles Peirce - Pragmatism, Charles Peirce - Scholastic realism, Charles Peirce - Formal perspective, Charles Peirce - Dynamics of representation, Charles Peirce - Normative sciences, Charles Peirce - Parallels with Leibniz, Charles Peirce - Bibliography, Charles Peirce - Primary literature, Charles Peirce - Secondary literature

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A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an element x there is no more than one element y such that x is related to y. This uniquely determined element y is denoted f(x). Because two definitions of binary relation are in use, there are actually two definitions of function, in ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Suppose P is any predicate in two variables that doesn't use the symbol B. Then in the formal language of the Zermelo-Fraenkel axioms, the axiom schema reads: or in words: If, given any set X, there is a unique set Y such that P holds for X and Y, then, given any set A, there is a set B such that, given any set C, C is a member of B if and only if there is a set D such that D is a member of A ...

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Axiom schema of replacement, Axiom schema of replacement - Statement, Axiom schema of replacement - Example applications, Axiom schema of replacement - History and philosophy, Axiom schema of replacement - Relation to the axiom schema of specification

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Objections to bundle theory concern the nature of the bundle of properties, the properties' compresence relation (the togetherness relation between those constituent properties), and the impact of language on understanding reality. Bundle theory - Compresence objection. Bundle theory maintains that properties are bundled together in a collection without describing how are they tied together. For example, bundle theory regards an apple as red, four inches (100 mm) wide, an ...

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Bundle theory, Bundle theory - Arguments for the bundle theory, Bundle theory - Objections to the bundle theory, Bundle theory - Compresence objection, Bundle theory - Language-reality objection, Bundle theory - Bundle Theory and Eastern Philosophy

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Determinacy - Games. The first sort of game we shall consider is the two-player game of perfect information of length ω, in which the players play natural numbers. In this sort of game we consider two players, often imaginatively named I and II, who take turns playing natural numbers, with I going first. They play "forever"; that is, their plays are indexed by the natural numbers. When they're finished, a predetermined condition decides which player won. This condition need not be specified by any definable rule; it may simply be an arbitrary lookup table saying who ...

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Determinacy, Determinacy - Basic notions, Determinacy - Games, Determinacy - Strategies, Determinacy - Winning strategies, Determinacy - Determined games, Determinacy - Determinacy from elementary considerations, Determinacy - Determinacy from ZFC, Determinacy - Determinacy and large cardinals, Determinacy - Measurable cardinals, Determinacy - Woodin cardinals, Determinacy - Projective determinacy, Determinacy - Axiom of determinacy, Determinacy - Consequences of determinacy, Determinacy - Regularity properties for sets of reals, Determinacy - Periodicity theorems, Determinacy - Properties of the Wadge hierarchy, Determinacy - More general games, Determinacy - Games in which the objects played are not natural numbers, Determinacy - Games played on trees, Determinacy - Long games, Determinacy - Games of imperfect information Blackwell games, Determinacy - Quasistrategies and quasideterminacy, Determinacy - Footnotes

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An antichain A of P is a subset such that if p and q are in A, then p and q are incompatible (written p ⊥ q), meaning there is no r in P such that r ≤ p and r ≤ q. In the Borel sets example, incompatibility means p∩q has measure zero. In the finite partial functions example, incompatibility means that p∪q is not a function, in other words ...

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Forcing mathematics, Forcing mathematics - Forcing posets, Forcing mathematics - Countable transitive models and generic filters, Forcing mathematics - Forcing, Forcing mathematics - Consistency, Forcing mathematics - Cohen forcing, Forcing mathematics - The countable chain condition, Forcing mathematics - Easton forcing, Forcing mathematics - Random reals, Forcing mathematics - Boolean-valued models, Forcing mathematics - Meta-mathematical explanation

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A string datatype is a datatype modeled on the idea of a formal string. Strings are such an important and useful datatype that they are implemented in nearly every programming language. In some languages they are available as primitive types and in others as composite types. The syntax of most high-level programming languages allows for a string, usually quoted in some way, to represent an instance of a string datatype; such a meta ...

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String computer science, String computer science - String datatypes, String computer science - Representations, String computer science - Memory management, String computer science - Vectors, String computer science - String algorithms, String computer science - Character string oriented languages and utilities, String computer science - Formal theory, String computer science - Character string functions

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Abstractions sometimes have ambiguous referents; for example, "happiness" (when used as an abstraction) can refer to many things as there are people and events or states of being which make them happy. A further example; suppose one attempts to define the term architecture and what it refers to. Architecture is more than simply designing safe functional buildings, it also involves elements of creation and innovation which aim at elegant solutions to problems of construction, the use of space, and at its best, to evoke an emotional response in the builders, owners, viewers and users of the building. See also:

Abstraction, Abstraction - Thought process, Abstraction - Conceptual schemes for abstraction, Abstraction - Referents, Abstraction - Instantiation, Abstraction - Physicality, Abstraction - Realness, Abstraction - Precise semantic meaning, Abstraction - Abstraction used in philosophy, Abstraction - Ontological status, Abstraction - Reification, Abstraction - Compression, Abstraction - The neurology of abstraction, Abstraction - Abstraction in Art

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Philosophers have many differing views on what the fundamental categories of being are. In no particular order, here are at least some items that have been regarded as categories of being by someone or other: Category of being - Physical objects. Physical objects are beings; certainly they are said to be in the simple sense that they exist all around us. So a house is a being, a person's body is a being, a tree is a being, a cloud is a being, and so on. They are beings because, and in ...

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Category of being, Category of being - Aristotle's Categories, Category of being - Categories of being, Category of being - Physical objects, Category of being - Minds, Category of being - Classes, Category of being - Properties, Category of being - Relations

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It is not sufficiently recognized that Peirce’s career was that of a scientist, not a philosopher; and that during his lifetime he was known and valued chiefly as a scientist, only secondly as a logician, and scarcely at all as a philosopher. Even his work in philosophy and logic will not be understood until this fact becomes a standing premise of Peircian studies. (Max Fisch, in Moore and Robin 1964: 486). Upon this first, and in one sense sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not ...

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Charles Peirce, Charles Peirce - Life, Charles Peirce - Reception, Charles Peirce - Works, Charles Peirce - Peirce's philosophy, Charles Peirce - Pragmatism, Charles Peirce - Scholastic realism, Charles Peirce - Formal perspective, Charles Peirce - Dynamics of representation, Charles Peirce - Normative sciences, Charles Peirce - Parallels with Leibniz, Charles Peirce - Bibliography, Charles Peirce - Primary literature, Charles Peirce - Secondary literature

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Many hypotheses in sciences such as physics can establish causality by noting that, until some phenomenon occurs, nothing happens; then when the phenomenon occurs, a second phenomenon is observed. But often in science, this situation is difficult to obtain. For example, in the old joke, someone claims that they are snapping their fingers "to keep the tigers away"; and justifies this behavior by saying "see - its working!" While this "experiment" does not falsify the hypothesis "snapping fingers keeps the tigers a ...

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Experiment, Experiment - An experiment in baking, Experiment - Design of experiments, Experiment - Controlled experiments, Experiment - Natural experiments, Experiment - Quasi-experiments, Experiment - Field Experiments, Experiment - Examples, Experiment - Quotes, Experiment - Literature

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If X = Y then we simply say that the binary relation is over X. Or it is an endorelation over X. Some important classes of binary relations over a set X are: reflexive: for all x in X it holds that xRx. For example, "greater than or equal to" is a reflexive relation but "greater than" is not. irreflexive: for all x in X it holds that not xRx. "Greater than" is an example of an irreflexive relation. See also:

Binary relation, Binary relation - Definition and examples, Binary relation - Definition, Binary relation - Remark, Binary relation - Example, Binary relation - Special types of relations, Binary relation - Relations over a set, Binary relation - Operations on binary relations, Binary relation - Examples of common binary relations

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As noted in the definition above two proportional variables are sometime said to be directly proportional. This is done so as to contrast proportionality with inverse proportionality. Two variables are inversely proportional (or varying inversely) if one of the variables is directly proportional with the multiplicative inverse of the other, or equivalently if their product is a constant. It follows, that the variable y is inversely proportional to the variable x if t ...

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Proportionality mathematics, Proportionality mathematics - Definition, Proportionality mathematics - Examples, Proportionality mathematics - Properties, Proportionality mathematics - Inverse proportionality, Proportionality mathematics - Exponential and logarithmic proportionality, Proportionality mathematics - Experimental determination

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A string datatype is a datatype modeled on the idea of a formal string. Strings are such an important and useful datatype that they are implemented in nearly every programming language. In some languages they are available as primitive types and in others as composite types. The syntax of most high-level programming languages allows for a string, usually quoted in some way, to represent an instance of a string datatype; such a meta ...

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String computer science, String computer science - String datatypes, String computer science - Representations, String computer science - Memory management, String computer science - String algorithms, String computer science - String oriented languages and utilities, String computer science - Formal theory

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Objections to bundle theory concern the nature of the bundle of properties, the properties' compresence relation (the togetherness relation between those constituent properties), and the impact of language on understanding reality. Bundle theory - Compressence objection. Bundle theory maintains that properties are bundled together in a collection without describing how are they tied together. For example, bundle theory regards an apple as red, four inches (100 mm) wide, a ...

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Bundle theory, Bundle theory - Arguments for the bundle theory, Bundle theory - Objections to the bundle theory, Bundle theory - Compressence objection, Bundle theory - Language-reality objection, Bundle theory - Bundle Theory and Eastern Philosophy

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For instance, starting with n = 6, one gets the sequence 6, 3, 10, 5, 16, 8, 4, 2, 1. (The Collatz conjecture says that this process always reaches 1, no matter what the starting value.) Starting with n = 11, the sequence takes longer to reach 1: 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. If the starting value n = 27 is chosen, the sequence takes 112 step ...

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Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

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A specific Collatz sequence can be easily computed, as is shown by this pseudocode example: def collatz(n) show n if n.odd? and n > 1 collatz(3n + 1) else if n.even? collatz(n / 2) This program halts when the sequence reaches 1, in order to avoid printing an endless cycle of 4, 2, 1. If the Collatz conjecture is true, the program will always halt no matter what positive starting integer is given to it. (See Halting problem for a discussion of the relation ...

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Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

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Consider the following operation on an arbitrary positive integer: If the number is even, divide it by two. If the number is odd, triple it and add one. For example, if this operation is performed on 3, the result is 10; if it is performed on 28, the result is 14. In mathematical notation, define the function f as follows: Now, form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as ...

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Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

Read more here: » Collatz conjecture: Encyclopedia II - Collatz conjecture - Statement of the problem