what do you mean by equivalence classes? explain with examples.

Asked by anchaka | 2nd Apr, 2013, 08:59: PM

Expert Answer:

In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element n in X is the subset of all elements in X which are equivalent to n. Equivalence classes among elements of a structure are often used to produce a smaller structure whose elements are the classes, distilling a relationship every element of the class shares with at least one other element of another class.
 
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 [7][9], and [1] 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.

Answered by  | 3rd Apr, 2013, 01:55: AM

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