- 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
.[3]
- The identity function (or identity map) from a set S to itself, often denoted
or
, such that
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
.[3]
Examples
Identity relation
A common example of the first meaning is the trigonometric identity
![{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9d143c522bbbe4226323295fda270d2ccf88e0)
which is true for all real values of
(since the real numbers
are the domain of both sine and cosine), as opposed to
![{\displaystyle \cos \theta =1,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ad57bae9c777475b1853399ba0e2d6c80c80774)
which is only true for certain values of
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
![{\displaystyle 0+a=a,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3f855d3d59f22cc2b043f05711b9c36073562a)
and
![{\displaystyle 0+0=0.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c88d7e69407adc76d018b1b2148d7b26e7e986)
Similarly, The number 1 is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all
![{\displaystyle 1\times a=a,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bde0d314d89660de5424261153f85181ee3dfea)
and
![{\displaystyle 1\times 1=1.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/217b0c74c307c9927a60d72d1f1541a5e0e2e2df)
Identity function
A common example of an identity function is the identity permutation, which sends each element of the set
to itself.
Comparison
These meanings are not mutually exclusive; for instance, the identity permutation is the identity element in the set of permutations of
under composition.