In mathematics, ∈-induction is a variant of transfinite induction, which can be used in set theory to prove that all sets satisfy a given property P(x). The induction hypothesis is for every set x, if ∀y (y∈x → P(y)), then P(x), from which we can infer that P(x) holds for all sets x. The ...

Read more here: » ∈-induction: Encyclopedia - ∈-induction

The concept of shattering of a set of points plays an important role in Vapnik Chervonenkis theory, also known as VC-theory. Shattering and VC-theory are used in the study of empirical processes as well as in statistical computational learning theory. Shattering - Definition. A class of sets, C, shatters a set A if and only if, for all there exists some such that , that is, if and only if For ...

Including:

• Shattering - Definition

Read more here: » Shattering: Encyclopedia - Shattering

Continuum can refer to: The continuous black body radiation spectrum. continuum, a "fretless keyboard" continuum, or mathematical sets (such as the real line) that can be contrasted with the properties of discrete spaces Continuum, a game client for the SubSpace computer game Continuum, a publishing house Continuum, an album by John Mayer Continuum, a musical composition by György Ligeti Continuum, a branch of physics that deals with continuous ...

Read more here: » Continuum: Encyclopedia - Continuum

Formally, we start with a category C with finite products (i.e. C has a terminal object 1 and any two objects of C have a product). A group object in C is an object G of C together with morphisms m : G × G → G (thought of as the "group multiplication") e : 1 → G (thought of as the "inclusion of the identity element") inv: G → G (thou ...

See also:

Group object, Group object - Definition, Group object - Examples, Group object - Group theory generalized

Read more here: » Group object: Encyclopedia II - Group object - Definition

If two equivalence relations over the set X are given, then their intersection (viewed as subsets of X×X) is also an equivalence relation. This allows for a convenient way of defining equivalence relations: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R. Concretely, the equivalence relation ~ generated by R can be described as follows: a ~ b if and only if there exist elements x_{See also:}

Equivalence relation, Equivalence relation - Examples of equivalence relations, Equivalence relation - Examples of relations that are not equivalences, Equivalence relation - Partitioning into equivalence classes, Equivalence relation - Generating equivalence relations, Equivalence relation - Common notions in Euclid's Elements

Read more here: » Equivalence relation: Encyclopedia II - Equivalence relation - Generating equivalence relations

Distributivity is most commonly found in rings and distributive lattices. A ring has two binary operations (commonly called "+" and "*"), and one of the requirements of a ring is that * must distribute over +. Most kinds of numbers (example 1) and matrices (example 3) form rings. A lattice is another kind of algebraic structure with two binary operations, ^ and v. If either of these operations (say ^) distributes over the other (v), then v must also distribute over ^, and the lattice is called distributive. See also the article ...

See also:

Distributivity, Distributivity - Definition, Distributivity - Examples, Distributivity - Generalizations of distributivity

Read more here: » Distributivity: Encyclopedia II - Distributivity - Examples

From one point of view, a groupoid is simply a category in which every morphism is an isomorphism (that is, invertible). To be explicit, a groupoid G is: A set G0 of objects; For each pair of objects x and y in G0, a set G(x,y) of morphisms (or arrows) from x to y — we write f : x → y to indicate that f is an elemen ...

See also:

Groupoid, Groupoid - Definitions, Groupoid - Examples, Groupoid - Relation to groups, Groupoid - Covariance in special relativity, Groupoid - Lie groupoids and Lie algebroids

Read more here: » Groupoid: Encyclopedia II - Groupoid - Definitions

A straight line is the shortest path between two points. Therefore the distance from A to B is no bigger than the length of the straight-line path from A to C plus the length of the straight-line path from C to B. This fact is called the triangle inequality. If that sum happens to be equal to the distance from A to B, then the three points A, B, and C lie on a straight ...

See also:

Distance geometry, Distance geometry - Introduction, Distance geometry - Cayley-Menger determinants

Read more here: » Distance geometry: Encyclopedia II - Distance geometry - Introduction

This is a complex question and, for simplicity of exposition, we will here consider only vacuous truth as concerns logical implication, i.e., the case when S has the form P ⇒ Q, and P is false. This case strikes many people as odd, and it's not immediately obvious whether all such statements are true, all such statements are false, some are true while others are false, or what. ...

See also:

Vacuous truth, Vacuous truth - Examples, Vacuous truth - Scope of the concept, Vacuous truth - Arguments of the semantic truth of vacuously true logical statements, Vacuous truth - Arguments that at least some vacuously true statements are true, Vacuous truth - Arguments for taking all vacuously true statements to be true, Vacuous truth - Arguments that only some vacuously true statements are true, Vacuous truth - Summary, Vacuous truth - Difficulties with the use of vacuous truth, Vacuous truth - Vacuous truths in mathematics

Read more here: » Vacuous truth: Encyclopedia II - Vacuous truth - Arguments of the semantic truth of vacuously true logical statements

Location filming has several advantages over filming on a studio set: It can be cheaper than constructing large sets The illusion of reality can be stronger - it is hard to replicate real-world wear-and-tear, and architectural details It's disadvantages include: A lack of control over the environment - passing aircraft, traffic, pedestrians, bad weather, city regulations, etc. Finding a real-world location which exactly matches the requirements of the script Taking a whole film crew to ...

See also:

Filming location, Filming location - Pros and cons, Filming location - Practicalities, Filming location - Substitute locations

Read more here: » Filming location: Encyclopedia II - Filming location - Pros and cons

In 2257, the United Planets Cruiser C-57D is sent to planet Altair IV, of the star-system Alpha Aquilae, to search for survivors of the ill-fated Bellerophon Expedition, lost some twenty years previously. As their ship arrives, the crew detects some immense power source scanning the ship. They are contacted by the sole survivor of the Expedition, Doctor Edward Morbius (Walter Pidgeon). Upon landing, they are met by Robby the Robot, who takes them to Morbius' home. Morbius explains that within a year of the Expedition arriving, some unknown f ...

See also:

Forbidden Planet, Forbidden Planet - Select Cast, Forbidden Planet - Plot, Forbidden Planet - About the film, Forbidden Planet - Soundtrack, Forbidden Planet - Influences on Pop Culture

Read more here: » Forbidden Planet: Encyclopedia II - Forbidden Planet - Plot

The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in those problems only arose much later. The doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (171 ...

See also:

Probability, Probability - Historical remarks, Probability - Concepts, Probability - Formalization of probability, Probability - Representation and interpretation of probability values, Probability - Distributions, Probability - Probability in mathematics, Probability - Remarks on probability calculations, Probability - Applications of probability theory to everyday life, Probability - Quotations

Read more here: » Probability: Encyclopedia II - Probability - Historical remarks

Cantor was born in St Petersburg, Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a Russian musician, Maria Anna Böhm. In 1856, the family moved to Germany and he continued his education in German schools, earning his doctorate from the University of Berlin in 1867. In 1890, he founded together with other mathematicians the Deutsche Mathematiker-Vereinigung and became the first president of the society. Cantor recognized that infinite sets can have different sizes, distinguished between countable and uncountab ...

See also:

Georg Cantor, Georg Cantor - Biography

Read more here: » Georg Cantor: Encyclopedia II - Georg Cantor - Biography

Mission sequence Mission 1: Arkhangelsk Dam: Byelomorye Dam Facility: Arkhangelsk Runway: Runway Mission 2: Servernaya Surface: Severnaya Bunker: Severnaya Mission 3: Kirghizstan Silo: Kirghizstan Mission 4: Monte Carlo Frigate: Frigate Mission ...

See also:

GoldenEye 007, GoldenEye 007 - Development, GoldenEye 007 - Gameplay and design, GoldenEye 007 - Storyline and missions, GoldenEye 007 - Additional missions, GoldenEye 007 - Multiplayer mode, GoldenEye 007 - Characters, GoldenEye 007 - Multiplayer Arenas, GoldenEye 007 - Weapons, GoldenEye 007 - Scenarios, GoldenEye 007 - Weapons, GoldenEye 007 - Easter eggs oddities and glitches, GoldenEye 007 - Unfinished features, GoldenEye 007 - The distant island, GoldenEye 007 - Citadel, GoldenEye 007 - All Bonds, GoldenEye 007 - Reaction, GoldenEye 007 - Sequels

Read more here: » GoldenEye 007: Encyclopedia II - GoldenEye 007 - Storyline and missions

Formally, two sets A and B are disjoint if their intersection is the empty set, i.e. if This definition extends to any collection of sets. A collection of sets is pairwise disjoint or mutually disjoint if any two distinct sets in the collection are disjoint. Formally, let I be an index set, and for each i in I, let Ai be a set. Then the family of sets {Ai : i ∈ I} is pairwise disjoint if for any i and j in I wit ...

See also:

Disjoint sets, Disjoint sets - Explanation

Read more here: » Disjoint sets: Encyclopedia II - Disjoint sets - Explanation

If two keys hash to the same index, the corresponding records cannot be stored in the same location. So, if it's already occupied, we must find another location where to store the new record, and do it so that we can find it when we look it up later on. To give an idea of the importance of a good collision resolution strategy, consider the following result, derived using the birthday paradox. Even if we assume that our hash function outputs random indices uniformly distributed over the array, and even for an array with 1 million entries, there is a 95% chance of at least one c ...

See also:

Hash table, Hash table - Overview, Hash table - Common uses of hash tables, Hash table - Choosing a good hash function, Hash table - Collision resolution, Hash table - Chaining, Hash table - Open addressing, Hash table - Open addressing versus chaining, Hash table - Coalesced hashing, Hash table - Perfect hashing, Hash table - Probabilistic hashing, Hash table - Table resizing, Hash table - Problems with hash tables, Hash table - Implementations

Read more here: » Hash table: Encyclopedia II - Hash table - Collision resolution

Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the set. One mathematical concept which closely models this naïve idea is that of topological dimension of a set. For example a point in the plane is described by two independent parameters (the cartesian coordinates of the point), so in this sense, the plane is two-dimensional. As one woul ...

See also:

Hausdorff dimension, Hausdorff dimension - Informal discussion, Hausdorff dimension - Formal definition, Hausdorff dimension - Results, Hausdorff dimension - Examples, Hausdorff dimension - Hausdorff dimension and topological dimension, Hausdorff dimension - Self-similar sets, Hausdorff dimension - Historical references

Read more here: » Hausdorff dimension: Encyclopedia II - Hausdorff dimension - Informal discussion

A relation over the sets X1, ..., Xn is an (n + 1)-tuple R=(X1, ..., Xn, G(R)) where G(R) is a subset of X1 × ... × Xn (the Cartesian product of these sets). If X=X1=X2=...=Xn, R is simply called a relation over X. G(R) is called the graph of R and, similar to the case of binary relations, R is often identified with its graph. An n-ary predi ...

See also:

Relation mathematics, Relation mathematics - Definition, Relation mathematics - Remarks

Read more here: » Relation mathematics: Encyclopedia II - Relation mathematics - Definition

There is no limit on movie length. The average movie length is between 30 seconds and 3 minutes, but user created films can be as long as budget will allow (although a 10 minute film might require over 20 game years of filming). Once the actors, extras, director and crew are assigned to a film, production begins, with the entire staff of the film travelling between sets to film the movie. The sandbox mode allows films to be created without undergoing the lengthy production process, and with budgets of over $100 million. See also:

The Movies, The Movies - General gameplay, The Movies - Staff Management, The Movies - Landscaping, The Movies - The Movies, The Movies - Scripts, The Movies - Post Production, The Movies - Genres, The Movies - Sets, The Movies - Technology, The Movies - Marketing, The Movies - Expandability and Customization, The Movies - Actors, The Movies - Crew, The Movies - Money, The Movies - File Sharing, The Movies - Costume and Painting

Read more here: » The Movies: Encyclopedia II - The Movies - The Movies

A function is bijective if it is both injective and surjective. A bijective function is a bijection (one-one correspondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. This equivalent condition is formally expressed as follows. The function is bijective iff for all , there is a unique such that f(a) = b. A function f : A → B is bijective if and only if it ...

See also:

Bijection injection and surjection, Bijection injection and surjection - Injection, Bijection injection and surjection - Surjection, Bijection injection and surjection - Bijection, Bijection injection and surjection - Cardinality, Bijection injection and surjection - Examples, Bijection injection and surjection - Injective and surjective bijective, Bijection injection and surjection - Injective and non-surjective, Bijection injection and surjection - Non-injective and surjective, Bijection injection and surjection - Non-injective and non-surjective, Bijection injection and surjection - Properties, Bijection injection and surjection - Category theory, Bijection injection and surjection - History

Read more here: » Bijection injection and surjection: Encyclopedia II - Bijection injection and surjection - Bijection

Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and two directions, infinite ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

Read more here: » Infinity: Encyclopedia II - Infinity - History

Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an initial object exists, then it is unique up to a unique isomorphism. The same is true for terminal objects. The automorphism group of an initial (or terminal) object I is trivial. Aut(I) = Hom(I,I) = { idI }. ...

See also:

Initial object, Initial object - Properties, Initial object - Examples

Read more here: » Initial object: Encyclopedia II - Initial object - Properties