Set (mathematics)A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics.
Family of setsIn set theory and related branches of mathematics, a collection of subsets of a given set is called a family of subsets of , or a family of sets over More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. A family of sets may be defined as a function from a set , known as the index set, to , in which case the sets of the family are indexed by members of .
Intersection (set theory)In set theory, the intersection of two sets and denoted by is the set containing all elements of that also belong to or equivalently, all elements of that also belong to Intersection is written using the symbol "" between the terms; that is, in infix notation. For example: The intersection of more than two sets (generalized intersection) can be written as: which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols.
Interval (mathematics)In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers , the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element).
Countably compact spaceIn mathematics a topological space is called countably compact if every countable open cover has a finite subcover. A topological space X is called countably compact if it satisfies any of the following equivalent conditions: (1) Every countable open cover of X has a finite subcover. (2) Every infinite set A in X has an ω-accumulation point in X. (3) Every sequence in X has an accumulation point in X. (4) Every countable family of closed subsets of X with an empty intersection has a finite subfamily with an empty intersection.
Compact discThe compact disc (CD) is a digital optical disc data storage format that was co-developed by Philips and Sony to store and play digital audio recordings. In August 1982, the first compact disc was manufactured. It was then released in October 1982 in Japan and branded as Digital Audio Compact Disc. It was released on March 2, 1983 in North America and Europe. The format was later adapted (as CD-ROM) for general-purpose data storage.
Unit intervalIn mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation I is most commonly reserved for the closed interval .
Boundary value problemIn the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems.
Sequentially compact spaceIn mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in . Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.
Robin boundary conditionIn mathematics, the Robin boundary condition (ˈrɒbɪn; properly ʁɔbɛ̃), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition.
Boundary (topology)In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include and . Some authors (for example Willard, in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds.
Cauchy boundary conditionIn mathematics, a Cauchy (koʃi) boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition. It is named after the prolific 19th-century French mathematical analyst Augustin-Louis Cauchy.
