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Unary Relation
In mathematics, a finitary relation over sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples consisting of elements ''x''''i'' in ''X''''i''. Typically, the relation describes a possible connection between the elements of an ''n''-tuple. For example, the relation "''x'' is divisible by ''y'' and ''z''" consists of the set of 3-tuples such that when substituted to ''x'', ''y'' and ''z'', respectively, make the sentence true. The non-negative integer ''n'' giving the number of "places" in the relation is called the ''arity'', ''adicity'' or ''degree'' of the relation. A relation with ''n'' "places" is variously called an ''n''-ary relation, an ''n''-adic relation or a relation of degree ''n''. Relations with a finite number of places are called ''finitary relations'' (or simply ''relations'' if the context is clear). It is also possible to generalize the concept to ''infinitary relations'' with infinite sequences. An ''n''-ary relation over sets is a ... [...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] |
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate ''T'' (for "tall") and assign it the extension (for "Abraham Lincoln"). Note that all our interpretation does is assign the extension to the non-logical constant ''T'', and does not make a claim about whether ''T'' is to stand for tall and ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
