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If ~ be an equivalence relation on a non-empty set A, then A can be expressed as a union of mutually disjoint equivalence classes in A

If ~ be an equivalence relation on a non-empty set A, then A can be expressed as a union of mutually disjoint equivalence classes in A

Grade:12th pass

1 Answers

Abhishek Singh
105 Points
2 years ago
Yes, This is a very fundamental Theorem in Set Theory. Expressing A as union of mutually disjoint classes is known as PARTITIONING the set.
 
Equivalence Class of an element a (denoted by [a]) is defined as the set of element equivalent to a.  
[a] = {x | x ~ a } 
 
Ex Let us define xRy (x is related to y ) if 2 divides x-y 
Now Checking Equivalence relation ship 
  1. Reflecxivity :  Clearly, x-x = 0 & 2 divides 0 
  2. Symmetery:  xRy ==> 2 divides x-y ==> 2 divides y-x ==> yRx
  3. Transitivity  xRy & yRz ==> 2 divides x-y & 2 divides y-z  ==> 2 divides [(x-y ) + (y-z)]=x-z ==> xRz 
Hence R is equivalence relation ship 
Checking the equivalence class we can see [0] = set of all even numbers  & [1] = set of all odd numbers
Thus the set integers is partitioned in  mututually disjoint (watertight) equivalence classes.
 
 

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