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:
“You say you are incapable of expressing your thought. How then do you explain the lucidity and brilliance with which you are expressing the thought that you are incapable of thought?”
—Jacques Rivière (1886–1925)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“The interest in life does not lie in what people do, nor even in their relations to each other, but largely in the power to communicate with a third party, antagonistic, enigmatic, yet perhaps persuadable, which one may call life in general.”
—Virginia Woolf (1882–1941)