Euclidean Relations
Euclid's The Elements includes the following "Common Notion 1":
- Things which equal the same thing also equal one another.
Nowadays, the property described by Common Notion 1 is called Euclidean (replacing "equal" by "are in relation with"). The following theorem connects Euclidean relations and equivalence relations:
Theorem. If a relation is Euclidean and reflexive, it is also symmetric and transitive.
Proof:
- (aRc ∧ bRc) → aRb = (aRa ∧ bRa) → aRb = bRa → aRb. Hence R is symmetric.
- (aRc ∧ bRc) → aRb = (aRc ∧ cRb) → aRb. Hence R is transitive.
Hence an equivalence relation is a relation that is Euclidean and reflexive. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed the reflexivity of equality too obvious to warrant explicit mention.
Read more about this topic: Equivalence Relation
Famous quotes containing the word relations:
“Major [William] McKinley visited me. He is on a stumping tour.... I criticized the bloody-shirt course of the canvass. It seems to me to be bad “politics,” and of no use.... It is a stale issue. An increasing number of people are interested in good relations with the South.... Two ways are open to succeed in the South: 1. A division of the white voters. 2. Education of the ignorant. Bloody-shirt utterances prevent division.”
—Rutherford Birchard Hayes (1822–1893)