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Quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of proper division). For example, when dividing 20 (the ''dividend'') by 3 (the ''divisor''), the ''quotient'' is "6 with a remainder of 2" in the Euclidean division sense, and 6\tfrac in the proper division sense. In the second sense, a quotient is simply the ratio of a dividend to its divisor. Notation The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line. The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole. \dfrac \quad \begin & \leftarrow \text \\ & \leftarrow \text \end \Biggr \} \leftarrow \text Integer part definition The q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space. Definition Let \left(X, \tau_X\right) be a topological space, and let \,\sim\, be an equivalence relation on X. The quotient set, Y = X / \sim\, is the set of equivalence clas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring and a two-sided ideal in , a new ring, the quotient ring , is constructed, whose elements are the cosets of in subject to special and operations. (Only the fraction slash "/" is used in quotient ring notation, not a horizontal fraction bar.) Quotient rings are distinct from the so-called "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization. Formal quotient ring construction Given a ring and a two-sided ideal in , we may define an equivalence relation on as follows: : if and only if is in . Using the ideal properties, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo ''n'' can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written G\,/\,N, where G is the original group and N is the normal subgroup. (This is pronounced G\bmod N, where \mbox is short for modulo.) Much of the importa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Space (linear Algebra)
In linear algebra, the quotient of a vector space ''V'' by a subspace ''N'' is a vector space obtained by "collapsing" ''N'' to zero. The space obtained is called a quotient space and is denoted ''V''/''N'' (read "''V'' mod ''N''" or "''V'' by ''N''"). Definition Formally, the construction is as follows. Let ''V'' be a vector space over a field ''K'', and let ''N'' be a subspace of ''V''. We define an equivalence relation ~ on ''V'' by stating that ''x'' ~ ''y'' if . That is, ''x'' is related to ''y'' if one can be obtained from the other by adding an element of ''N''. From this definition, one can deduce that any element of ''N'' is related to the zero vector; more precisely, all the vectors in ''N'' get mapped into the equivalence class of the zero vector. The equivalence class – or, in this case, the coset – of ''x'' is often denoted : 'x''= ''x'' + ''N'' since it is given by : 'x''= . The quotient space ''V''/''N'' is then defined as ''V''/~, the set of all equival ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \,\sim\, on S, the of an element a in S, denoted by is the set \ of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by \,\sim\,, and is denoted by S / \sim. When the set S has some structure (such as a group operation or a topology) and the equivalence relation \,\sim\, is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Set
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \,\sim\, on S, the of an element a in S, denoted by is the set \ of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by \,\sim\,, and is denoted by S / \sim. When the set S has some structure (such as a group operation or a topology) and the equivalence relation \,\sim\, is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Category
In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting. Definition Let ''C'' be a category. A '' congruence relation'' ''R'' on ''C'' is given by: for each pair of objects ''X'', ''Y'' in ''C'', an equivalence relation ''R''''X'',''Y'' on Hom(''X'',''Y''), such that the equivalence relations respect composition of morphisms. That is, if :f_1,f_2 : X \to Y\, are related in Hom(''X'', ''Y'') and :g_1,g_2 : Y \to Z\, are related in Hom(''Y'', ''Z''), then ''g''1''f''1 and ''g''2''f''2 are related in Hom(''X'', ''Z''). Given a congruence relation ''R'' on ''C'' we can define the quotient category ''C''/''R'' as the category whose objects are those of ''C'' and whose morphisms are equivalence classes of morphisms in ''C''. That is, :\mathrm_(X,Y) = \mathrm_(X,Y)/R_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Graph
In graph theory, a quotient graph ''Q'' of a graph ''G'' is a graph whose vertices are blocks of a partition of the vertices of ''G'' and where block ''B'' is adjacent to block ''C'' if some vertex in ''B'' is adjacent to some vertex in ''C'' with respect to the edge set of ''G''. In other words, if ''G'' has edge set ''E'' and vertex set ''V'' and ''R'' is the equivalence relation induced by the partition, then the quotient graph has vertex set ''V''/''R'' and edge set . More formally, a quotient graph is a quotient object in the category of graphs. The category of graphs is concretizable – mapping a graph to its set of vertices makes it a concrete category – so its objects can be regarded as "sets with additional structure", and a quotient graph corresponds to the graph induced on the quotient set ''V''/''R'' of its vertex set ''V''. Further, there is a graph homomorphism (a quotient map) from a graph to a quotient graph, sending each vertex or edge to the equivalence class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Type
In type theory, a kind of foundation of mathematics, a quotient type is an algebraic data type that represents a type whose equality relation has been redefined by a given equivalence relation such that the elements of the type are partitioned into a set of equivalence classes whose cardinality is less than or equal to that of the base type. Just as product types and sum types are analogous to the cartesian product and disjoint sum of abstract algebraic structures, quotient types reflect the concept of set-theoretic quotients, sets whose elements are surjectively partitioned into equivalence classes by a given equivalence relation on the set. Algebraic structures whose underlying set is a quotient are also termed quotients. Examples of such quotient structures include quotient sets, groups, rings, categories and, in topology, quotient spaces. In type theories that lack quotient types, setoids – sets explicitly equipped with an equivalence relation – are often used inst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division (mathematics)
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication. At an elementary level the division of two natural numbers is, among other possible interpretations, the process of calculating the number of times one number is contained within another. This number of times need not be an integer. For example, if 20 apples are divided evenly between 4 people, everyone receives 5 apples (see picture). The division with remainder or Euclidean division of two natural numbers provides an integer ''quotient'', which is the number of times the second number is completely contained in the first number, and a ''remainder'', which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. For example, if 21 apples are divided between 4 people, everyone receives 5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division (mathematics)
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication. At an elementary level the division of two natural numbers is, among other possible interpretations, the process of calculating the number of times one number is contained within another. This number of times need not be an integer. For example, if 20 apples are divided evenly between 4 people, everyone receives 5 apples (see picture). The division with remainder or Euclidean division of two natural numbers provides an integer ''quotient'', which is the number of times the second number is completely contained in the first number, and a ''remainder'', which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. For example, if 21 apples are divided between 4 people, everyone receives 5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
