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Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.Contents1 Definition 2 Automorphism group 3 Examples 4 History 5 Inner and outer automorphisms 6 See also 7 References 8 External linksDefinition[edit] In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator). The identity morphism (identity mapping) is called the trivial automorphism in some contexts
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Mathematics
Mathematics
Mathematics
(from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist
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Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.[1] An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix. The definition of kernel takes various forms in various contexts. But in all of them, the kernel of a homomorphism is trivial (in a sense relevant to that context) if and only if the homomorphism is injective
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Associativity
In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider the following equations: ( 2 + 3 ) + 4 = 2 + ( 3 + 4 ) = 9 displaystyle (2+3)+4=2+(3+4)=9, 2 × ( 3 × 4 ) = ( 2 × 3 ) × 4 = 24. displaystyle 2times (3times 4)=(2times 3)times 4=24
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Identity Element
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them. This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion, however, the identity implicitly depends on the binary operation it is coupled with. Let (S, ∗) be a set S with a binary operation ∗ on it. Then an element e of S is called a left identity if e ∗ a = a for all a in S, and a right identity if a ∗ e = a for all a in S
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Set Theory
Set theory
Set theory
is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor
Georg Cantor
and Richard Dedekind
Richard Dedekind
in the 1870s
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Permutation
In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting. These differ from combinations, which are selections of some members of a set where order is disregarded. For example, written as tuples, there are six permutations of the set 1,2,3 , namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this three element set. As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters. In this example, the letters are already ordered in the original word and the anagram is a reordering of the letters. The study of permutations of finite sets is a topic in the field of combinatorics. Permutations occur, in more or less prominent ways, in almost every area of mathematics
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Elementary Arithmetic
Elementary arithmetic
Elementary arithmetic
is the simplified portion of arithmetic that includes the operations of addition, subtraction, multiplication, and division. It should not be confused with elementary function arithmetic. Elementary arithmetic
Elementary arithmetic
starts with the natural numbers and the written symbols (digits) that represent them
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Integer
An integer (from the Latin
Latin
integer meaning "whole")[note 1] is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, ​5 1⁄2, and √2 are not. The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, …), also called whole numbers or counting numbers,[1][2] and their additive inverses (the negative integers, i.e., −1, −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold Z displaystyle mathbb Z ( Unicode
Unicode
U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers").[3][4] Z is a subset of the set of all rational numbers Q, in turn a subset of the real numbers R. Like the natural numbers, Z is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers
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Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain. Evaluating a function at each element of a subset X of the domain, produces a set called the image of X under or through the function. The inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S. Image and inverse image may also be defined for general binary relations, not just functions.Contents1 Definition1.1 Image of an element 1.2 Image of a subset 1.3 Image of a function2 Inverse image 3 Notation for image and inverse image3.1 Arrow notation 3.2 Star notation 3.3 Other terminology4 Examples 5 Consequences 6 See also 7 Notes 8 ReferencesDefinition[edit] The word "image" is used in three related ways
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Trivial Group
In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0, 1 or e depending on the context
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Function Composition
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function. For instance, the functions f : X → Y and g : Y → Z can be composed to yield a function which maps x in X to g(f(x)) in Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.[note 1] The notation g ∘ f is read as "g circle f ", "g round f ", "g about f ", "g composed with f ", "g after f ", "g following f ", "g of f", or "g on f "
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Linear Algebra
Linear
Linear
algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , displaystyle a_ 1 x_ 1 +cdots +a_ n x_ n =b, linear functions such as ( x 1 , … , x n ) ↦ a 1 x 1 + … + a n x n , displaystyle (x_ 1 ,ldots ,x_ n )mapsto a_ 1 x_ 1 +ldots +a_ n x_ n , and their representations through matrices and vector spaces.[1][2][3] Linear
Linear
algebra is central to almost all areas of mathematics
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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. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements
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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. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics. The best known fields are the field of rational numbers and the field of real numbers. The field of complex numbers is also widely used, not only in mathematics, but also in many areas of science and engineering. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension
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Rational Number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold Q displaystyle mathbb Q , Unicode ℚ);[2] it was thus denoted in 1895 by Giuseppe Peano
Giuseppe Peano
after quoziente, Italian for "quotient". The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g
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