Expressing Properties of Binary Relations in RA
The following table shows how many of the usual properties of binary relations can be expressed as succinct RA equalities or inequalities. Below, an inequality of the form A≤B is shorthand for the Boolean equation A∨B = B.
The most complete set of results of this nature is chpt. C of Carnap (1958), where the notation is rather distant from that of this entry. Chpt. 3.2 of Suppes (1960) contains fewer results, presented as ZFC theorems and using a notation that more resembles that of this entry. Neither Carnap nor Suppes formulated their results using the RA of this entry, or in an equational manner.
| R is | If and only if: |
|---|---|
| Functional | R•R ≤ I |
| Total or Connected | I ≤ R•R (R is surjective) |
| Function | functional and total. |
| Injective |
R•R ≤ I (R is functional) |
| Surjective | I ≤ R•R (R is total) |
| Bijection | R•R = R•R = I (Injective surjective function) |
| Reflexive | I ≤ R |
| Coreflexive | R ≤ I |
| Irreflexive | R ∧ I = 0 |
| Transitive | R•R ≤ R |
| Preorder | R is reflexive and transitive. |
| Antisymmetric | R ∧ R ≤ I |
| Partial order | R is an antisymmetric preorder. |
| Total order | R is a total partial order. |
| Strict partial order | R is transitive and irreflexive. |
| Strict total order | R is a total strict partial order. |
| Symmetric | R = R |
| Equivalence | R•R = R. R is a symmetric preorder. |
| Asymmetric | R ≠ R |
| Dense | R ∧ I– ≤ (R ∧ I–)•(R ∧ I–). |
Read more about this topic: Relation Algebra
Famous quotes containing the words expressing, properties and/or relations:
“What means the fact—which is so common, so universal—that some soul that has lost all hope for itself can inspire in another listening soul an infinite confidence in it, even while it is expressing its despair?”
—Henry David Thoreau (1817–1862)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“I know all those people. I have friendly, social, and criminal relations with the whole lot of them.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)