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Encyclopedia > 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. It leaves other elements unchanged when combined with them. This is used for groups and related concepts. In philosophy, identity has both metaphysical and logical senses. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...


The term identity element is often shortened to identity when there is no possibility of confusion; we do so in this article.


Let (S,*) be a set S with a binary operation * on it (known as a magma). 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. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity. In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...


An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). The distinction is used most often for sets that support both binary operations, such as rings. The multiplicative identity is often called the unit in the latter context, where, unfortunately, a unit is also sometimes used to mean an element with a multiplicative inverse. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... 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

set operation identity
real numbers + (addition) 0
real numbers • (multiplication) 1
real numbers ab (exponentiation) 1 (right identity only)
m-by-n matrices + (addition) zero matrix
n-by-n square matrices • (multiplication) identity matrix
all functions from a set M to itself ∘ (function composition) identity map
all functions from a set M to itself * (convolution) δ (Dirac delta)
character strings, lists concatenation empty string, empty list
extended real numbers minimum/infimum +∞
extended real numbers maximum/supremum -∞
subsets of a set M ∩ (intersection) M
sets ∪ (union) { } (empty set)
boolean logic ∧ (logical and) ⊤ (truth)
boolean logic ∨ (logical or) ⊥ (falsity)
compact surfaces # (connected sum) S2
only two elements {e, f} * defined by
e * e = f * e = e and
f * f = e * f = f
both e and f are left identities, but there is no right or two-sided identity

As the last example shows, it is possible for (S,*) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if l is a left identity and r is a right identity then l = l * r = r. In particular, there can never be more than one two-sided identity. If there were two, e and f, then e * f would have to be equal to both e and f. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... For other senses of this word, see zero or 0. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article is about the number one. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, a zero matrix is a matrix with all its entries being zero. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ... In various branches of mathematics and computer science, strings are sequences of various simple objects (symbols, tokens, characters, etc. ... In various branches of mathematics and computer science, strings are sequences of various simple objects (symbols, tokens, characters, etc. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... Boolean logic is a complete system for logical operations. ... Boolean logic is a complete system for logical operations. ...


See also


  Results from FactBites:
 
Identity element - definition of Identity element in Encyclopedia (284 words)
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 term identity element is often shortened to identity when there is no possibility of confusion; we do so in this article.
But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity.
Math Forum - Ask Dr. Math (414 words)
For example, for multiplication, the identity element is 1, because a * 1 = a For addition, the identity element is 0, because a + 0 = a For matrix multiplication, the identity element is _ _
The equation must be true for all elements b in the given set for e to be declared the identity element of the set.
If the binary operation is addition, then the identity is zero since 0 + a = a + 0 = a However if the binary operation is multiplication, then the identity is 1 since: 1 * a = a * 1 = a I hope the idea is clear.
  More results at FactBites »

 
 

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