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The element 0 is usually referred to as the identity element and if it exists, it is unique
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In mathematics, a function that always returns the same value that was used as its argument
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to be this identity element. Such a definition generalizes to the concept of an
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726:. Mathematical Association of America Textbooks. Mathematical Assn of Amer.
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773:
A First Course in
Topology: An Introduction to Mathematical Thinking
603:...then the diagonal set determined by M is the identity relation...
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458:
142:
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Conferences, University of
Michigan Engineering Summer (1968).
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we see that an identity element of a semigroup is idempotent.
30:
362:, one can alternately define the identity function on
256:, where a function is defined as a particular kind of
649:
591:. American Mathematical Society. 1974. p. 92.
68:that always returns the value that was used as its
446:(essentially multiplication by 1), considered in
814:
701:. Undergraduate Texts in Mathematics. Springer.
682:Elementary Linear Algebra (Applications Version)
565:(11th ed.). Sarat Book House. p. 36.
800:Foundations of Information Systems Engineering
650:Rosales, J. C.; García-Sánchez, P. A. (1999).
473:containing only this isometry (symmetry type
694:
417:the identity function is represented by the
796:
722:D. Marshall; E. Odell; M. Starbird (2007).
588:Proceedings of Symposia in Pure Mathematics
698:Applied Linear Algebra and Matrix Analysis
358:Since the identity element of a monoid is
260:, the identity function is given by the
193:is always the same as the input element
29:
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277:
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653:Finitely Generated Commutative Monoids
457:the identity function is trivially an
438:The identity function on the positive
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776:. Courier Corporation. p. 65.
684:(9th ed.), Wiley International
614:
561:Mapa, Sadhan Kumar (7 April 2014).
176:In other words, the function value
13:
563:Higher Algebra Abstract and Linear
487:, the identity function is always
444:completely multiplicative function
14:
854:
770:Conover, Robert A. (2014-05-21).
131:is defined to be a function with
37:of the identity function on the
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656:. Nova Publishers. p. 1.
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355:(under function composition).
76:is the identity function, the
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724:Number Theory through Inquiry
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161: for all elements
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833:Basic concepts in set theory
7:
502:
395:The identity function is a
205:. The identity function on
72:, unchanged. That is, when
10:
859:
537:Knapp, Anthony W. (2006),
217:(its codomain is also its
93:is true for all values of
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627:10.1007/978-3-319-31159-3
494:The identity function is
461:. An object without any
119:, the identity function
19:Not to be confused with
386:need not be functions.
62:identity transformation
828:Elementary mathematics
823:Functions and mappings
680:Anton, Howard (2005),
343:of all functions from
296:is any function, then
228:The identity function
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695:T. S. Shores (2007).
435:chosen for the space.
33:
431:, regardless of the
324:function composition
322:, where "∘" denotes
278:Algebraic properties
240:is often denoted by
751:Hyperbolic Geometry
615:Nel, Louis (2016).
215:surjective function
838:Types of functions
519:Indicator function
211:injective function
42:
783:978-0-486-78001-6
753:, Springer 2005,
747:James W. Anderson
708:978-038-733-195-9
663:978-1-56072-670-8
636:978-3-319-31159-3
618:Continuity Theory
598:978-0-8218-1425-3
572:978-93-80663-24-1
548:978-0-8176-3248-9
485:topological space
370:identity morphism
326:. In particular,
262:identity relation
54:identity relation
52:, also called an
50:identity function
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509:Identity matrix
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621:. p. 21.
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209:is clearly an
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25:Empty function
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539:Basic algebra
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213:as well as a
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109:Formally, if
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21:Null function
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541:, Springer,
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455:metric space
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376:, where the
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58:identity map
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43:
39:real numbers
465:has as its
412:dimensional
46:mathematics
843:1 (number)
817:Categories
525:References
496:idempotent
489:continuous
390:Properties
254:set theory
105:Definition
223:bijective
97:to which
503:See also
463:symmetry
459:isometry
440:integers
287: :
266:diagonal
143:codomain
78:equality
70:argument
66:function
339:of the
335:is the
137:as its
64:, is a
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757:
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705:
660:
633:
595:
569:
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406:In an
360:unique
341:monoid
139:domain
483:In a
453:In a
442:is a
433:basis
264:, or
219:range
115:is a
48:, an
35:Graph
778:ISBN
755:ISBN
728:ISBN
703:ISBN
658:ISBN
631:ISBN
593:ISBN
567:ISBN
543:ISBN
469:the
311:= id
301:∘ id
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141:and
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623:doi
380:of
372:in
349:to
282:If
268:of
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117:set
60:or
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23:or
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328:id
317:∘
307:=
291:→
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242:id
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477:1
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427:n
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