Identity (mathematics)

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.

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 .
The identity function (or identity map) from a set S to itself, often denoted  or , such that  for all x in S.
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 .

 Examples 
Identity relation
A common example of the first meaning is the trigonometric identity

which is true for all real values of  (since the real numbers  are the domain of both sine and cosine), as opposed to

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 

 and

Similarly, The number 1 is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all 

 and

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.

 Related pages 

 Identity Property
 One
 Zero

 References 

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

Category:Algebra