Please explain the theory of Equivalence Class in relations.

Dear ria,

Equivalence relation, a mathematical concept, is a type of relation on a given set that provides a way for elements of that set to be identified with (meaning considered equivalent to for some present purpose) other elements of that set. Equivalence relation is defined in a branch of mathematics called set theory, a vital branch underpinning all branches of mathematics and those fields that use mathematics. The power of an equivalence relation lies in its ability to partition a set into the disjoint union of subsets called equivalence classes. Because of its power to partition a set, an equivalence relation is one of the most used and pervasive tools in mathematics. 

Given a set, what can a mathematician do with it? The mathematician can consider the elements of it. For example, the mathematician can consider the elements of the set of non-negative integers: {0, 1, 2 …}. Moreover, in this example (call it "clockwork" arithmetic), suppose the mathematician wants to consider time measured in hours. How can the mathematician express the idea that 12 is equivalent to, or as mathematicians also say identified with, 24 or 48? In general, take any non-negative integer. Divide this integer by 12 and keep the remainder. Any non-negative number that gives the same remainder in this way is equivalent to any other such number. Hence, 2 is equivalent to 14, 26, 38 …. By positing that any non-negative number is equivalent to its remainder after division by 12, the mathematician succeeds in gaining a new perspective on the set of non-negative integers. The definition of equivalence relation is based on this simple idea of considering some elements to be equivalent to others under the equivalence relation.

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