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. In a group (an algebraic structure in mathematics), the identity element is sometimes denoted by the symbol .[1][2][3]
Further Examples
change| 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
changeReferences
change- ↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-13.
- ↑ Weisstein, Eric W. "Identity Element". mathworld.wolfram.com. Retrieved 2020-08-13.
- ↑ "Definition of IDENTITY ELEMENT". www.merriam-webster.com. Retrieved 2020-08-13.
