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Single Idea 10104

[filed under theme 4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes ]

Full Idea

A relation is an equivalence relation if it is reflexive, symmetric and transitive. The 'same first letter' is an equivalence relation on the set of English words. Any relation that puts a partition into clusters will be equivalence - and vice versa.

Gist of Idea

'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words

Source

A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)

Book Ref

George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.64

A Reaction

This is a key concept in the Fregean strategy for defining numbers.

The 7 ideas with the same theme [classes created by close relationships of members]:

9854 We can introduce new objects, as equivalence classes of objects already known [Frege, by Dummett]
9883 Frege introduced the standard device, of defining logical objects with equivalence classes [Frege, by Dummett]
9715 An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton]
9716 We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton]
8920 Equivalence relations are reflexive, symmetric and transitive, and classify similar objects [Lipschutz]
10104 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman]
10103 Grouping by property is common in mathematics, usually using equivalence [George/Velleman]