EQUIVALENCE RELATION

equivalence relation: translation

(or equivalence class )

A relation which is transitive, symmetric, and reflexive divides its field into exclusive classes of things. Within each class everything bears the relation to everything else, and nothing bears it to anything in a different class. ‘Of the same height as’ or ‘having the same number of members as’ are equivalence relations that divide the field of people into classes of equally tall people, and sets into different classes depending on how many members they have. Defining an equivalence class of things at the same point on the scale is the fundamental operation in establishing any kind of measure of a property or quantity. In mathematical logic an early definition of a cardinal number ( Frege's ) was as an equivalence class of equinumerous classes.