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Mathematical Object A MATHEMATICAL OBJECT is an abstract object arising in mathematics . The concept is studied in philosophy of mathematics . In mathematical practice, an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs . Commonly encountered mathematical objects include numbers , permutations , partitions , matrices , sets , functions , and relations . Geometry Geometry as a branch of mathematics has such objects as hexagons , points , lines , triangles , circles , spheres , polyhedra , topological spaces and manifolds . Another branch—algebra —has groups , rings , fields , grouptheoretic lattices , and ordertheoretic lattices . Categories are simultaneously homes to mathematical objects and mathematical objects in their own right. In proof theory , proofs and theorems are also mathematical objects [...More...]  "Mathematical Object" on: Wikipedia Yahoo 

Additive Inverse In mathematics, the ADDITIVE INVERSE of a number a is the number that, when added to a, yields zero . This number is also known as the OPPOSITE (number), SIGN CHANGE, and NEGATION. For a real number , it reverses its sign : the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself. The additive inverse of a is denoted by unary minus : −a (see the discussion below ). For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0 . The additive inverse is defined as its inverse element under the binary operation of addition (see the discussion below ), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect : −(−x) = x [...More...]  "Additive Inverse" on: Wikipedia Yahoo 

Addition ADDITION (often signified by the plus symbol "+") is one of the four basic operations of arithmetic , with the others being subtraction , multiplication and division . The addition of two whole numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two apples together, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" i.e., "3 add 2 is equal to 5". Besides counting fruits, addition can also represent combining other physical objects. Using systematic generalizations, addition can also be defined on more abstract quantities, such as integers , rational numbers , real numbers and complex numbers and other abstract objects such as vectors and matrices . In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others. In algebra, addition is studied more abstractly [...More...]  "Addition" on: Wikipedia Yahoo 

Additive Identity In mathematics the ADDITIVE IDENTITY of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics , but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings . CONTENTS * 1 Elementary examples * 2 Formal definition * 3 Further examples * 4 Proofs * 4.1 The additive identity is unique in a group * 4.2 The additive identity annihilates ring elements * 4.3 The additive and multiplicative identities are different in a nontrivial ring * 5 See also * 6 References * 7 External links ELEMENTARY EXAMPLES * The additive identity familiar from elementary mathematics is zero, denoted 0 [...More...]  "Additive Identity" on: Wikipedia Yahoo 

Essence In philosophy , ESSENCE is the property or set of properties that make an entity or substance what it fundamentally is, and which it has by necessity , and without which it loses its identity . Essence is contrasted with accident : a property that the entity or substance has contingently , without which the substance can still retain its identity. The concept originates with Aristotle, who used the Greek expression to ti ên einai (τὸ τί ἦν εἶναι, literally meaning "the what it was to be" and corresponding to the scholastic term quiddity ) or sometimes the shorter phrase to ti esti (τὸ τί ἐστι, literally meaning "the what it is" and corresponding to the scholastic term haecceity ) for the same idea. This phrase presented such difficulties for its Latin translators that they coined the word essentia (English "essence") to represent the whole expression [...More...]  "Essence" on: Wikipedia Yahoo 

Paradox A PARADOX is a statement that, despite apparently sound reasoning from true premises, leads to a selfcontradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time. Some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking . Some paradoxes have revealed errors in definitions assumed to be rigorous, and have caused axioms of mathematics and logic to be reexamined. One example is Russell\'s paradox , which questions whether a "list of all lists that do not contain themselves" would include itself, and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed. Others, such as Curry\'s paradox , are not yet resolved [...More...]  "Paradox" on: Wikipedia Yahoo 

Operation (mathematics) In mathematics , an OPERATION is a calculation from zero or more input values (called operands ) to an output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations of arity 2, such as addition and multiplication , and unary operations of arity 1, such as additive inverse and multiplicative inverse . An operation of arity zero, or 0ary operation is a constant . The mixed product is an example of an operation of arity 3, or ternary operation . Generally, the arity is supposed to be finite, but infinitary operations are sometimes considered. In this context, the usual operations, of finite arity are also called FINITARY OPERATIONS [...More...]  "Operation (mathematics)" on: Wikipedia Yahoo 

Georg Cantor GEORG FERDINAND LUDWIG PHILIPP CANTOR (/ˈkæntɔːr/ KANtor ; German: ; March 3 1845 – January 6, 1918 ) was a German mathematician. He invented set theory , which has become a fundamental theory in mathematics. Cantor established the importance of onetoone correspondence between the members of two sets, defined infinite and wellordered sets , and proved that the real numbers are more numerous than the natural numbers . In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware. Cantor's theory of transfinite numbers was originally regarded as so counterintuitive – even shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J [...More...]  "Georg Cantor" on: Wikipedia Yahoo 

Lattice (group) In geometry and group theory , a LATTICE in R n {displaystyle mathbb {R} ^{n}} is a subgroup of R n {displaystyle mathbb {R} ^{n}} which is isomorphic to Z n {displaystyle mathbb {Z} ^{n}} , and which spans the real vector space R n {displaystyle mathbb {R} ^{n}} . In other words, for any basis of R n {displaystyle mathbb {R} ^{n}} , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell . Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras , number theory and group theory. They also arise in applied mathematics in connection with coding theory , in cryptography because of conjectured computational hardness of several lattice problems , and are used in various ways in the physical sciences [...More...]  "Lattice (group)" on: Wikipedia Yahoo 

Lattice (order) A LATTICE is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra . It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join ) and a unique infimum (also called a greatest lower bound or meet ). An example is given by the natural numbers , partially ordered by divisibility , for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities . Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra . Semilattices include lattices, which in turn include Heyting and Boolean algebras . These "latticelike" structures all admit ordertheoretic as well as algebraic descriptions [...More...]  "Lattice (order)" on: Wikipedia Yahoo 

Theorem In mathematics , a THEOREM is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms . A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system . The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive , in contrast to the notion of a scientific law , which is experimental . Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called HYPOTHESES or premises [...More...]  "Theorem" on: Wikipedia Yahoo 

Ontology ONTOLOGY is the philosophical study of the nature of being , becoming , existence and/or reality , as well as the basic categories of being and their relations. Traditionally listed as a part of the major branch of philosophy known as metaphysics , ontology often deals with questions concerning what entities exist or may be said to exist and how such entities may be grouped, related within a hierarchy , and subdivided according to similarities and differences. Although ontology as a philosophical enterprise is highly hypothetical, it also has practical application in information science and technology , such as ontology engineering . A very simple definition of ontology is that it is the examination of what is meant, in context, by the word 'thing' [...More...]  "Ontology" on: Wikipedia Yahoo 

Foundations Of Mathematics FOUNDATIONS OF MATHEMATICS is the study of the philosophical and logical and/or algorithmic basis of mathematics , or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical figure, set, function, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts , with an eye to the philosophical aspects and the unity of mathematics [...More...]  "Foundations Of Mathematics" on: Wikipedia Yahoo 

Model Theory In mathematics , MODEL THEORY is the study of classes of mathematical structures (e.g. groups , fields , graphs , universes of set theory ) from the perspective of mathematical logic . The objects of study are models of theories in a formal language . A set of sentences in a formal language is called a THEORY; a MODEL of a theory is a structure (e.g. an interpretation ) that satisfies the sentences of that theory. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang ">', but also to construct free algebras [...More...]  "Model Theory" on: Wikipedia Yahoo 

Graph (discrete Mathematics) In mathematics , and more specifically in graph theory , a GRAPH is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called an arc or line). Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics . The edges may be directed or undirected [...More...]  "Graph (discrete Mathematics)" on: Wikipedia Yahoo 

Vertex (geometry) In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place [...More...]  "Vertex (geometry)" on: Wikipedia Yahoo 