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
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