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Relation Construction
In logic and mathematics, relation construction and relational constructibility have to do with the ways that one relation is determined by an indexed family or a sequence of other relations, called the ''relation dataset''. The relation in the focus of consideration is called the ''faciendum''. The relation dataset typically consists of a specified relation over sets of relations, called the ''constructor'', the ''factor'', or the ''method of construction'', plus a specified set of other relations, called the ''faciens'', the ''ingredients'', or the ''makings''. Relation composition and relation reduction are special cases of relation constructions. See also * Projection * Relation * Relation composition In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplic ... {{DEFAULTSORT:Relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation (mathematics)
In mathematics, a relation on a set may, or may not, hold between two given set members. For example, ''"is less than"'' is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1 is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : for all , or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : every nonempty subset of contains a minimal element with respect to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indexed Family
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, where a given function selects one real number for each integer (possibly the same). More formally, an indexed family is a mathematical function together with its domain I and image X. (that is, indexed families and mathematical functions are technically identical, just point of views are different.) Often the elements of the set X are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set I is called the ''index set'' of the family, and X is the ''indexed set''. Sequences are one type of families indexed by natural numbers. In general, the index set I is not restricted to be countable. For example, one could consider an uncountable family of sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation Composition
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplication, and its result is called a relative product. Function composition is the special case of composition of relations where all relations involved are functions. The word uncle indicates a compound relation: for a person to be an uncle, he must be the brother of a parent. In algebraic logic it is said that the relation of Uncle (x U z) is the composition of relations "is a brother of" (x B y) and "is a parent of" (y P z). U = BP \quad \text \quad xByPz \text xUz. Beginning with Augustus De Morgan, the traditional form of reasoning by syllogism has been subsumed by relational logical expressions and their composition. Definition If R \subseteq X \times Y and S \subseteq Y \times Z are two binary relations, then their composition R; S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (mathematics)
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a projection is also called a ''projection'', even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: * The projection from a point onto a plane or central projection: If ''C'' is a point, called the center of projection, then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation (mathematics)
In mathematics, a relation on a set may, or may not, hold between two given set members. For example, ''"is less than"'' is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1 is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : for all , or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : every nonempty subset of contains a minimal element with respect to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
