Identity element

In mathematics, the identity element (or neutral element) of a set is a special element of that set. It is special because if it is combined with another element of that set, it does not change the other element.

With addition, the identity element is 0, because adding 0 to some number does not change the number. With multiplication, it is 1. In a group (an algebraic structure in mathematics), the identity element is sometimes denoted by the symbol ${\displaystyle e}$.^[1][2][3]

Further Examples

set	operation	identity
real numbers	+ (addition)	0
real numbers	• (multiplication)	1
real numbers	ab (exponentiation)	1 (right identity only)
m-by-n matrices	+ (addition)	zero matrix
n-by-n square matrices	• (multiplication)	identity matrix
all functions from a set M to itself	∘ (function composition)	identity map
character strings, lists	concatenation	empty string, empty list
extended real numbers	minimum/infimum	+∞
extended real numbers	maximum/supremum	-∞
subsets of a set M	∩ (intersection)	M
sets	∪ (union)	{} (empty set)
boolean logic	∧ (logical and)	⊤ (truth)
boolean logic	∨ (logical or)	⊥ (falsity)
only two elements {e, f}	* defined by e * e = f * e = e and f * f = e * f = f	both e and f are left identities, but there is no right or two-sided identity

Related pages

• Identity (mathematics)
• Null vector
• One
• Zero