Identity function

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Graph of the identity function on the real numbers

In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f(x) = x.

Definition[edit]

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

f(x) = x   for all elements x in M.[1]

In other words, the function value f(x) in M (that is, the codomain) is always the same input element x of M (now considered as the domain). The identity function on M is clearly an injective function as well as a surjective function, so it is also bijective.[2]

The identity function f on M is often denoted by idM.

In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of M.

Algebraic property[edit]

If f : MN is any function, then we have f ∘ idM = f = idNf (where "∘" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M.

Since the identity element of a monoid is unique, one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.

Properties[edit]

See also[edit]

References[edit]

  1. ^ Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 978-0-8176-3248-9 
  2. ^ Mapa, Sadhan Kumar. Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36. ISBN 978-93-80663-24-1. 
  3. ^ Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International 
  4. ^ D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519. 
  5. ^ T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 038-733-195-6. 
  6. ^ James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1-85233-934-9