# Identity element

special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them

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.

## Further Examples

set operation identity
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