Equivalence Class - Examples

Examples

  • If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. X/~ could be naturally identified with the set of all car colors (cardinality of X/~ would be the number of all car colors)
  • Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their difference x − y is an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation, and all represent the same element of Z/~.
  • Let X be the set of ordered pairs of integers (a,b) with b not zero, and define an equivalence relation ~ on X according to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b) can be identified with the rational number a/b, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers. The same construction can be generalized to the field of fractions of any integral domain.

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