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Isomorphism

Index Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. [1]

110 relations: Abstract algebra, Alexandroff extension, Algebra, Algebraic structure, American football, Ancient Greek, Antisymmetric relation, Archie Manning, Asymmetric relation, Automorphism, Bertrand Russell, Bijection, Binary relation, Bisimulation, Canonical map, Cardinality, Category (mathematics), Category theory, Cauchy sequence, Chinese remainder theorem, Class (set theory), Commutative diagram, Complex conjugate, Concrete category, Continuous function, Coprime integers, Cybernetics, Cyclic group, Dedekind cut, Diffeomorphism, Differentiable function, Differential equation, Dimension (vector space), Direct product of groups, Dual space, Eli Manning, Equality (mathematics), Equivalence class, Equivalence relation, Exponential function, Extensional and intensional definitions, Field (mathematics), Genealogy, Good regulator, Graph isomorphism, Graph theory, Greatest and least elements, Group (mathematics), Group homomorphism, Group isomorphism, ..., Heap (mathematics), Hilbert space, Homeomorphism, Homology (mathematics), Homomorphism, Introduction to Mathematical Philosophy, Inverse function, Isometry, Isomorphism class, Isomorphism theorems, John F. Kennedy, Joseph P. Kennedy Sr., Laplace transform, Linear map, List of small groups, Logarithm, Logical atomism, Ludwig Wittgenstein, Map (mathematics), Mathematical analysis, Mathematical object, Mathematical table, Mathematics, Modular arithmetic, Morphism, Natural transformation, Order isomorphism, Order theory, Partially ordered set, Peyton Manning, Positive real numbers, Principal homogeneous space, Projective line, Quarterback, Quotient space (topology), Real number, Reflexive relation, Riemann sphere, Ring (mathematics), Ring homomorphism, Robert F. Kennedy, Root of unity, Row and column vectors, Ruler, Serial relation, Set-builder notation, Slide rule, Subquotient, Symmetric group, Symmetric relation, Topological space, Topology, Total order, Transitive relation, Transpose, Universal property, Variety (universal algebra), Vector space, Vertex (graph theory), Weak ordering. Expand index (60 more) »

Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

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Alexandroff extension

In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact.

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Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

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Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.

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American football

American football, referred to as football in the United States and Canada and also known as gridiron, is a team sport played by two teams of eleven players on a rectangular field with goalposts at each end.

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Ancient Greek

The Ancient Greek language includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD.

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Antisymmetric relation

In mathematics, a binary relation R on a set X is anti-symmetric if there is no pair of distinct elements of X each of which is related by R to the other.

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Archie Manning

Elisha Archibald Manning III (born May 19, 1949) is a former American football quarterback who played professionally for 13 seasons in the National Football League (NFL).

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Asymmetric relation

In mathematics, an asymmetric relation is a binary relation on a set X where.

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Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

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Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, historian, writer, social critic, political activist, and Nobel laureate.

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Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

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Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

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Bisimulation

In theoretical computer science a bisimulation is a binary relation between state transition systems, associating systems that behave in the same way in the sense that one system simulates the other and vice versa.

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Canonical map

In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects.

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Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

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Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

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Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

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Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

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Chinese remainder theorem

The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer by several integers, then one can determine uniquely the remainder of the division of by the product of these integers, under the condition that the divisors are pairwise coprime.

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Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

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Commutative diagram

The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.

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Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

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Concrete category

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, see Relative concreteness below).

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

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Cybernetics

Cybernetics is a transdisciplinary approach for exploring regulatory systems—their structures, constraints, and possibilities.

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Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

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Dedekind cut

In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind, are а method of construction of the real numbers from the rational numbers.

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Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

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Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

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Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

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Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

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Direct product of groups

In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.

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Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

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Eli Manning

Elisha Nelson Manning IV (born January 3, 1981) is an American football quarterback for the New York Giants of the National Football League (NFL).

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Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.

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Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

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Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

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Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

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Extensional and intensional definitions

Extensional and intensional definitions are two key ways in which the object(s) or concept(s) a term refers to can be defined.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Genealogy

Genealogy (from γενεαλογία from γενεά, "generation" and λόγος, "knowledge"), also known as family history, is the study of families and the tracing of their lineages and history.

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Good regulator

The good regulator is a theorem conceived by Roger C. Conant and W. Ross Ashby that is central to cybernetics.

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Graph isomorphism

In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H such that any two vertices u and v of G are adjacent in G if and only if ƒ(u) and ƒ(v) are adjacent in H. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection.

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Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

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Greatest and least elements

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Group homomorphism

In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".

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Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.

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Heap (mathematics)

In abstract algebra, a heap (sometimes also called a groud) is a mathematical generalization of a group.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

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Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

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Homomorphism

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

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Introduction to Mathematical Philosophy

Introduction to Mathematical Philosophy is a book by Bertrand Russell, published in 1919, written in part to exposit in a less technical way the main ideas of his and Whitehead's Principia Mathematica (1910–13), including the theory of descriptions.

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Inverse function

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.

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Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

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Isomorphism class

An isomorphism class is a collection of mathematical objects isomorphic to each other.

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Isomorphism theorems

In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects.

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John F. Kennedy

John Fitzgerald "Jack" Kennedy (May 29, 1917 – November 22, 1963), commonly referred to by his initials JFK, was an American politician who served as the 35th President of the United States from January 1961 until his assassination in November 1963.

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Joseph P. Kennedy Sr.

Joseph Patrick Kennedy Sr. (September 6, 1888 – November 18, 1969) was an American businessman, investor, and politician known for his high-profile positions in United States politics.

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Laplace transform

In mathematics, the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

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Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

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Logical atomism

Logical atomism is a philosophical belief that originated in the early 20th century with the development of analytic philosophy.

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Ludwig Wittgenstein

Ludwig Josef Johann Wittgenstein (26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language.

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Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Mathematical object

A mathematical object is an abstract object arising in mathematics.

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Mathematical table

Mathematical tables are lists of numbers showing the results of calculation with varying arguments.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

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Morphism

In mathematics, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.

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Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.

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Order isomorphism

In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets).

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Order theory

Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.

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Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

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Peyton Manning

Peyton Williams Manning (born March 24, 1976) is a former American football quarterback who played 18 seasons in the National Football League (NFL), primarily with the Indianapolis Colts.

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Positive real numbers

In mathematics, the set of positive real numbers, \mathbb_.

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Principal homogeneous space

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial.

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Projective line

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.

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Quarterback

A quarterback (commonly abbreviated "QB") is a position in American and Canadian football.

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Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Reflexive relation

In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.

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Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

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Robert F. Kennedy

Robert Francis "Bobby" Kennedy (November 20, 1925 – June 6, 1968) was an American politician and lawyer who served as the 64th United States Attorney General from January 1961 to September 1964, and as a U.S. Senator for New York from January 1965 until his assassination in June 1968.

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Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

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Row and column vectors

In linear algebra, a column vector or column matrix is an m × 1 matrix, that is, a matrix consisting of a single column of m elements, Similarly, a row vector or row matrix is a 1 × m matrix, that is, a matrix consisting of a single row of m elements Throughout, boldface is used for the row and column vectors.

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Ruler

A ruler, sometimes called a rule or line gauge, is a device with equally spaced markings along its length, used in geometry, technical drawing, engineering and building to measure distances or to rule straight lines.

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Serial relation

In set theory, a serial relation is a binary relation R for which every element of the domain has a corresponding range element (∀ x ∃ y x R y).

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Set-builder notation

In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy.

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Slide rule

The slide rule, also known colloquially in the United States as a slipstick, is a mechanical analog computer.

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Subquotient

In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject.

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Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.

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Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a. In mathematical notation, this is: Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

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Transitive relation

In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.

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Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

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Universal property

In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem.

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Variety (universal algebra)

In the mathematical subject of universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Vertex (graph theory)

In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).

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Weak ordering

In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other.

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References

[1] https://en.wikipedia.org/wiki/Isomorphism

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