What is the Partition of Set-- Mathematics

Let, “S” be a non-empty set. A collection of P={A_1,A_2,A_3… } of non-empty subsets of “S” is called a partition of “S”, if
Some conditions are applied on the subsets of the set “S”;
A_1∪A_2∪A_3∪…∪A_n=S
If, A_i≠A_j where i≠j
A_i∩A_j=∅,disjoint sets ∴i and j=1,2,3,…n
For instance; let, I={…,-3,-2,-1,0,1,2,3…}, then
The collection of{A_1,A_2,A_3},
Where A_1={1,2,3,4,…}, A_2={0} and A_3={-1,-2,-3,-4,…}
Are the nonempty and disjoint subsets of set I, then write all the subsets in a single set called P.
Now, check for all conditions
Condition 1st:
A_i∪A_j=S
So,
A_1∪A_2∪A_3=I
I={1,2,3,4,…}∪{0}∪{-1,-2,-3,-4,…}
I={…,-3,-2,-1,0,1,2,3…}
It is satisfies the first condition.
Condition 2nd:
A_i≠A_j where i≠j
A_1≠A_2≠A_3={1,2,3,4,…}≠{0}≠{-1,-2,-3,-4,…}
It is satisfies the second condition.
Condition 3rd:
A_i∩A_j=∅,disjoint sets ∴i and j=1,2,3,…n
A_1∩A_2∩A_3=∅
A_1∩A_2∩A_3={1,2,3,4,…}∩{0}∩{-1,-2,-3,-4,…}
A_1∩A_2∩A_3=∅
It is satisfies the third condition.
Now collect all the nonempty subsets in set P
That is; P=[{1,2,3,4,…},{0},{-1,-2,-3,-4,…}] Therefore the set P is a partition of set I.
Note: it may be seen that every equilance relation on asset determines a unique partition of the set and every partition of a set defines an equilance relation on the set.
Let,
P={{1},{2},{3},{4}}
R={(1,1),(2,2),(3,3),(4,4)}
Example:
Let, A={1,2,3,4} and consider the partition P={{1,2},{3,4}} of A, find the relation R on A determined by P. Sets in a disjoint set is relation to every other element in the same disjoint set and only to those elements. Thus,
R={(1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(4,3),(4,4)}
R=A×A