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Identity (mathematics)




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For other senses of this word, see identity.

In mathematics, the term identity has several important uses:

  • An identity is an equality that remains true even if you change all the variables that are used in that equality.[1][2]

An equality in mathematical sense is only true under more particular conditions. For this, the symbol ≡ is sometimes used (note, however, that the same symbol can also be used for a congruence relation as well.)

  • In algebra, an identity or identity element of a set S with an operation is an element which, when combined with any element s of S, produces s itself. In a group (an algebraic structure), this is often denoted by the symbol [math]e[/math].[3]
  • The identity function (or identity map) from a set S to itself, often denoted [math]\mathrm{id}[/math] or [math]\mathrm{id}_S[/math], such that [math]\mathrm{id}(x)=x[/math] for all x in S.[4]
  • 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. It is often denoted by the symbol [math]I[/math].[3]

Examples

Identity relation

A common example of the first meaning is the trigonometric identity

[math] \sin ^2 \theta + \cos ^2 \theta = 1\,[/math]

which is true for all real values of [math]\theta[/math] (since the real numbers [math]\Bbb{R}[/math] are the domain of both sine and cosine), as opposed to

[math]\cos \theta = 1,\,[/math]

which is only true for certain values of [math]\theta[/math] in a subset of the domain.

Identity element

The concepts of "additive identity" and "multiplicative identity" are central to the Peano axioms. The number 0 is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all [math]a\in\Bbb{R},[/math]

[math]0 + a = a,\,[/math]
[math]a + 0 = a,\,[/math] and
[math]0 + 0 = 0.\,[/math]

Similarly, The number 1 is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all [math]a\in\Bbb{R},[/math]

[math]1 \times a = a,\,[/math]
[math]a \times 1 = a,\,[/math] and
[math]1 \times 1 = 1.\,[/math]

Identity function

A common example of an identity function is the identity permutation, which sends each element of the set [math]\{ 1, 2, \ldots, n \}[/math] to itself.

Comparison

These meanings are not mutually exclusive; for instance, the identity permutation is the identity element in the set of permutations of [math]\{ 1, 2, \ldots, n \}[/math] under composition.

Related pages

References

  1. "The Definitive Glossary of Higher Mathematical Jargon" (in en-US). 2019-08-01. https://mathvault.ca/math-glossary/. 
  2. "Identity - Math Open Reference". https://www.mathopenref.com/identity.html. 
  3. 3.0 3.1 "Comprehensive List of Algebra Symbols" (in en-US). 2020-03-25. https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/. 
  4. Weisstein, Eric W.. "Identity Map" (in en). https://mathworld.wolfram.com/IdentityMap.html. 

Other websites

  • EquationSolver - A webpage that can test a suggested identity and return a true/false "verdict".