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Set Of Sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I, known as the index set, to F, in which case the sets of the family are indexed by members of I. In some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class. A finite family of subsets of a finite set S is also called a ''hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the power set o ... [...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] |
Transversal (combinatorics)
In mathematics, particularly in combinatorics, given a family of sets, here called a collection ''C'', a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of ''C'' (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal: * One variation is that there is a bijection ''f'' from the transversal to ''C'' such that ''x'' is an element of ''f''(''x'') for each ''x'' in the transversal. In this case, the transversal is also called a system of distinct representatives (SDR). * The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of ''C''. In this situation, the members of the system of representatives are not necessarily distinct. In computer science, comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology (structure)
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error-correcting Code
In computing, telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The central idea is that the sender encodes the message in a redundant way, most often by using an error correction code, or error correcting code (ECC). The redundancy allows the receiver not only to detect errors that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a reverse channel to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth. The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Distance
In information theory, the Hamming distance between two String (computer science), strings or vectors of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to change one string into the other, or equivalently, the minimum number of ''errors'' that could have transformed one string into the other. In a more general context, the Hamming distance is one of several string metrics for measuring the edit distance between two sequences. It is named after the American mathematician Richard Hamming. A major application is in coding theory, more specifically to block codes, in which the equal-length strings are Vector space, vectors over a finite field. Definition The Hamming distance between two equal-length strings of symbols is the number of positions at which the corresponding symbols are different. Examples The symbols may be letters, bits, or decimal digits, am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String (computer Science)
In computer programming, a string is traditionally a sequence of character (computing), characters, either as a literal (computer programming), literal constant or as some kind of Variable (computer science), variable. The latter may allow its elements to be Immutable object, mutated and the length changed, or it may be fixed (after creation). A string is often implemented as an array data structure of bytes (or word (computer architecture), words) that stores a sequence of elements, typically characters, using some character encoding. More general, ''string'' may also denote a sequence (or List (abstract data type), list) of data other than just characters. Depending on the programming language and precise data type used, a variable (programming), variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements. When a string appears lit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Code
In coding theory, block codes are a large and important family of Channel coding, error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, mathematicians, and computer scientists to study the limitations of ''all'' block codes in a unified way. Such limitations often take the form of ''bounds'' that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors. Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay code (other), Golay codes, Reed–Muller codes and Polar code (coding theory), Polar codes. These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x is an element of X and y is an element of Y. It encodes the common concept of relation: an element x is ''related'' to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime p is related to each integer z that is a Divisibility, multiple of p, but not to an integer that is not a Multiple (mathematics), multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as -4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Binary relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the Point (geometry), points and Line (geometry), lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the heterogeneous relation, relation of which points are incident (geometry), incident on which lines for all points and lines. What is left is the incidence structure of the Euclidean plane. Incidence structures are most often considered in the geometrical context where they are abstracted from, and hence generalize, planes (such as affine plane (incidence geometry), affine, projective plane, projective, and Möbius planes), but the concept is very broad and not limited to geometric settings. Even in a geometric setting, incidence structures are not limited to just points and lines; higher-dimensional objects (Plane (mathematics), planes, Solid geometry, sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Complex
In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersections of the elements are also included in the set (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely Combinatorics, combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex., Section 4.3 Definitions A simplicial complex \mathcal is a set of Simplex, simplices that satisfies the following conditions: # Every Simplex#Elements, face of a simplex from \mathcal is also in \mathcal. # The non-empty Set intersection, intersection of any two simplices \sigma_1, \sigma_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
