Examples
| set | operation | identity |
|---|---|---|
| real numbers | + (addition) | 0 |
| real numbers | · (multiplication) | 1 |
| non-negative numbers | ab (exponentiation) | 1 (right identity only) * |
| integers (to extended rationals) | ||
| positive integers | least common multiple | 1 |
| non-negative integers | greatest common divisor | 0 (under most definitions of GCD) |
| m-by-n matrices | + (addition) | matrix of all zeroes |
| n-by-n square matrices | matrix multiplication | In (matrix with 1 on diagonal and 0 elsewhere) |
| m-by-n matrices | (Hadamard product) | Jm, n (Matrix of ones) |
| all functions from a set M to itself | ∘ (function composition) | identity function |
| all distributions on an group G | ∗ (convolution) | δ (Dirac delta) |
| 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) |
| a boolean algebra | ∧ (logical and) | ⊤ (truth) |
| a boolean algebra | ∨ (logical or) | ⊥ (falsity) |
| a boolean algebra | ⊕ (Exclusive or) | ⊥ (falsity) |
| knots | knot sum | unknot |
| compact surfaces | # (connected sum) | S2 |
| 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 identity and no two-sided identity |
* assuming 00 = 1, or zero has to be excluded from the domain.
Read more about this topic: Identity Element
Famous quotes containing the word examples:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
Main Site Subjects