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.
Cartesian productIn mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. In terms of set-builder notation, that is A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).
Computer graphicsComputer graphics deals with generating s and art with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. A great deal of specialized hardware and software has been developed, with the displays of most devices being driven by computer graphics hardware. It is a vast and recently developed area of computer science. The phrase was coined in 1960 by computer graphics researchers Verne Hudson and William Fetter of Boeing.
Graphics processing unitA graphics processing unit (GPU) is a specialized electronic circuit initially designed to accelerate computer graphics and (either on a video card or embedded on the motherboards, mobile phones, personal computers, workstations, and game consoles). After their initial design, GPUs were found to be useful for non-graphic calculations involving embarrassingly parallel problems due to their parallel structure. Other non-graphical uses include the training of neural networks and cryptocurrency mining.
SubsetIn mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements. The subset relation defines a partial order on sets.
Graphics cardA graphics card (also called a video card, display card, graphics adapter, VGA card/VGA, video adapter, display adapter, or colloquially GPU) is a computer expansion card that generates a feed of graphics output to a display device such as a monitor. Graphics cards are sometimes called discrete or dedicated graphics cards to emphasize their distinction to integrated graphics processor on the motherboard or the CPU.
Notation for differentiationIn differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.
Leibniz's notationIn calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively. Consider y as a function of a variable x, or y = f(x).
Graphics Core NextGraphics Core Next (GCN) is the codename for a series of microarchitectures and an instruction set architecture that were developed by AMD for its GPUs as the successor to its TeraScale microarchitecture. The first product featuring GCN was launched on January 9, 2012. GCN is a reduced instruction set SIMD microarchitecture contrasting the very long instruction word SIMD architecture of TeraScale. GCN requires considerably more transistors than TeraScale, but offers advantages for general-purpose GPU (GPGPU) computation due to a simpler compiler.
Vector graphicsVector graphics is a form of computer graphics in which visual images are created directly from geometric shapes defined on a Cartesian plane, such as points, lines, curves and polygons. The associated mechanisms may include vector display and printing hardware, vector data models and file formats, as well as the software based on these data models (especially graphic design software, computer-aided design, and geographic information systems).
2D computer graphics2D computer graphics is the generation of s—mostly from two-dimensional models (such as 2D geometric models, text, and digital images) and by techniques specific to them. It may refer to the branch of computer science that comprises such techniques or to the models themselves. 2D computer graphics are mainly used in applications that were originally developed upon traditional printing and drawing technologies, such as typography, cartography, technical drawing, advertising, etc.
Direct productIn mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the , which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.
Raster graphicsIn computer graphics and digital photography, a raster graphics represents a two-dimensional picture as a rectangular matrix or grid of square pixels, viewable via a computer display, paper, or other display medium. A raster is technically characterized by the width and height of the image in pixels and by the number of bits per pixel. Raster images are stored in s with varying dissemination, production, generation, and . The printing and prepress industries know raster graphics as contones (from continuous tones).
Open setIn mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set along with a distance defined between any two points), an open set is a set that, along with every point P, contains all points that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P). More generally, an open set is a member of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself.
Meagre setIn the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre.
Rendering (computer graphics)Rendering or image synthesis is the process of generating a photorealistic or non-photorealistic image from a 2D or 3D model by means of a computer program. The resulting image is referred to as the render. Multiple models can be defined in a scene file containing objects in a strictly defined language or data structure. The scene file contains geometry, viewpoint, texture, lighting, and shading information describing the virtual scene. The data contained in the scene file is then passed to a rendering program to be processed and output to a or raster graphics image file.
Disjoint setsIn mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. This definition of disjoint sets can be extended to families of sets and to indexed families of sets.
DefinitionA definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.
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 .
Mathematical notationMathematical notation consists of using symbols for representing operations, unspecified numbers, relations, and any other mathematical objects and assembling them into expressions and formulas. Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. For example, Albert Einstein's equation is the quantitative representation in mathematical notation of the mass–energy equivalence.
