Abhishek Singh
Last Activity: 3 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
- Reflecxivity : Clearly, x-x = 0 & 2 divides 0
- Symmetery: xRy ==> 2 divides x-y ==> 2 divides y-x ==> yRx
- 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.
