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Identity function

From Wikipedia, the free encyclopedia
Function that returns its argument unchanged
Not to be confused withNull function orEmpty function.
Graph of the identity function on thereal numbers

Inmathematics, anidentity function, also called anidentity relation,identity map oridentity transformation, is afunction that always returns the value that was used as itsargument, unchanged. That is, whenf is the identity function, theequalityf(x) =x is true for all values ofx to whichf can be applied.

Definition

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Formally, ifX is aset, the identity functionf onX is defined to be a function withX as itsdomain andcodomain, satisfying

f(x) =x for all elementsx inX.[1]

In other words, the function valuef(x) in the codomainX is always the same as the input elementx in the domainX. The identity function onX is clearly aninjective function as well as asurjective function (its codomain is also itsrange), so it isbijective.[2]

The identity functionf onX is often denoted byidX.

Inset theory, where a function is defined as a particular kind ofbinary relation, the identity function is given by theidentity relation, ordiagonal ofX.[3]

Algebraic properties

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Iff :X →Y is any function, thenf ∘ idX =f = idY ∘f, where "∘" denotesfunction composition.[4] In particular,idX is theidentity element of themonoid of all functions fromX toX (under function composition).

Since the identity element of a monoid isunique,[5] one can alternately define the identity function onM to be this identity element. Such a definition generalizes to the concept of anidentity morphism incategory theory, where theendomorphisms ofM need not be functions.

Properties

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See also

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References

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  1. ^Knapp, Anthony W. (2006).Basic algebra. Springer.ISBN 978-0-8176-3248-9.
  2. ^Mapa, Sadhan Kumar (7 April 2014).Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36.ISBN 978-93-80663-24-1.
  3. ^Proceedings of Symposia in Pure Mathematics. American Mathematical Society. 1974. p. 92.ISBN 978-0-8218-1425-3....then the diagonal set determined by M is the identity relation...
  4. ^Nel, Louis (2016).Continuity Theory. Cham: Springer. p. 21.doi:10.1007/978-3-319-31159-3.ISBN 978-3-319-31159-3.
  5. ^Rosales, J. C.; García-Sánchez, P. A. (1999).Finitely Generated Commutative Monoids. Nova Publishers. p. 1.ISBN 978-1-56072-670-8.The element 0 is usually referred to as the identity element and if it exists, it is unique
  6. ^Anton, Howard (2005),Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
  7. ^T. S. Shores (2007).Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer.ISBN 978-038-733-195-9.
  8. ^D. Marshall; E. Odell; M. Starbird (2007).Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer.ISBN 978-0883857519.
  9. ^Anderson, James W. (2007).Hyperbolic geometry. Springer undergraduate mathematics series (2. ed., corr. print ed.). London: Springer.ISBN 978-1-85233-934-0.
  10. ^Conover, Robert A. (2014-05-21).A First Course in Topology: An Introduction to Mathematical Thinking. Courier Corporation. p. 65.ISBN 978-0-486-78001-6.
  11. ^Conferences, University of Michigan Engineering Summer (1968).Foundations of Information Systems Engineering.we see that an identity element of a semigroup is idempotent.
Types by domain and codomain
Classes/properties
Constructions
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