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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] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions and is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: : a_0 = a_1 = 1 and a_n = a_ + (n - 1)a_ for n > 1. The first few terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ISO 31-11
ISO 31-11:1992 was the part of international standard ISO 31 that defines ''mathematical signs and symbols for use in physical sciences and technology''. It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-80000-2:2019. It included definitions for symbols for mathematical logic, set theory, arithmetic and complex numbers, functions and special functions and values, matrices, vectors, and tensors, coordinate systems, and miscellaneous mathematical relations. See also * Mathematical symbols * 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 ... References and notes {{Mathematical symbols notation language Mathematical symbols Mathematical notation #00031-11 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Compliment
Relative may refer to: General use *Kinship and family, the principle binding the most basic social units of society. If two people are connected by circumstances of birth, they are said to be ''relatives''. Philosophy * Relativism, the concept that points of view have no absolute truth or validity, having only relative, subjective value according to differences in perception and consideration, or relatively, as in the relative value of an object to a person * Relative value (philosophy) Economics * Relative value (economics) Popular culture Film and television * ''Relatively Speaking'' (1965 play), 1965 British play * ''Relatively Speaking'' (game show), late 1980s television game show * ''Everything's Relative'' (episode)#Yu-Gi-Oh! (Yu-Gi-Oh! Duel Monsters), 2000 Japanese anime ''Yu-Gi-Oh! Duel Monsters'' episode *'' Relative Values'', 2000 film based on the play of the same name. *'' It's All Relative'', 2003-4 comedy television series *''Intelligence is Relative'', tag li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Setminus
In set theory, the complement of a set , often denoted by A^c (or ), is the set of elements not in . When all elements in the universe, i.e. all elements under consideration, are considered to be 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 : A^c= U \setminus A = \. The absolute complement of is usually denoted by A^c. Other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
