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In mathematics, the inverse of an element x, with respect to an operation *, is an element x' such that their compose gives a neutral element. This generalizes the concepts of opposite and reciprocal of a number, and inverse functions, among others. An element is invertible iff it has an inverse. Math sucks. ...
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. ...
The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...
In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ...
In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...
In mathematics, the inverse of an element x, with respect to an operation *, is a chicken fucker, x such that their compose gives a neutral element. ...
↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...
Introduction
The idea of inverse element generalises the concepts of (arithmetic) negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element. The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...
Addition (or summation) is one of the basic operations of arithmetic. ...
The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...
In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ...
In its simplest form, multiplication is a quick way of adding identical numbers. ...
Formal definition Let S be a set with a binary operation *. If e is an identity element of (S,*) and a * b = e, then a is called a left inverse of b and b is called a right inverse of a. If an element x is both a left inverse and a right inverse of y, then x is called a two-sided inverse, or simply an inverse, of y. An element with a two-sided inverse in S is called invertible in S. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ...
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. ...
Just like (S,*) can have several left identities or several right identities, it is possible for an element to have several left inverses or several right inverses (but note that their definition above uses a two-sided identity e). It can even have several left inverses and several right inverses.
However if the operation is associative, then if an element has both a left inverse and a right inverse, then they are equal and unique. In this case, the set of (left and right) invertible elements is a group, called the group of units of S, and denoted by U(S) or S*. In mathematics, associativity is a property that a binary operation can have. ...
The term group can refer to several concepts: Look up Group on Wiktionary, the free dictionary In music, a group is another term for band or other musical ensemble. ...
In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...
Examples In addition to the opposite (− x) and reciprocal (1/x) of numbers, an important example is the notion of an invertible square matrix: An n×n matrix M over a field K is invertible if and only if its determinant is different from zero. If the determinant of M is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one. For the square matrix section, see square matrix. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
More generally, a square matrix over a ring K is invertible iff its determinant is invertible in K. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...
↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...
A function g is the left (resp. right) inverse of a function f (for function composition 'o'), iff g o f (resp. f o g) is the identity function on the domain (resp. codomain) of f. In this example, it is very frequent for a function to have a right inverse and no left inverse, or the converse. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...
Domain has several meanings (see subsections below for computer-related and mathematical senses): some kind of territory, such as (for example) a demesne or a realm In New Zealand a Town Domain is typically a public sport area administered by a Domain Board. ...
A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ...
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