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```				   Please explain the theory of Equivalence Class in relations.
```

6 years ago

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```										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.

We are all  IITians and here to help you in your IIT  JEE preparation.  All the best.

Sagar Singh
B.Tech IIT Delhi

```
6 years ago

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