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Disjoint Sets
Post: #1

Disjoint Sets

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Equivalence Relations

A relation R is defined on a set S if for every pair of elements (a,b), a,bS, a R b is either true or false. If a R b is true, then we say that a is related to b.
An equivalence relation is a relation R that satisfy three properties:
(reflexive) a R a, for all a  S.
(symmetric) a R b if and only if b R a.
(transitive) a R b and b R c implies that a R c.

Disjoint Sets

Make the input a collection of N sets, each with one element.
All relations (except reflexive) are false;
Each set has a different element: SiSj= => it makes the sets disjoint.
Find operation: returns the name of the set containing a given element.
Add/Union operation (e.g., add relation a~b)
Check if a and b are already related: if they are in the same equivalence class.
If not, apply “union”: merge the two equivalence classes containing a and b into a new equivalence class.


On a set of subsets, the three operations amount to:
Create a set of n disjoint subsets with every node in its own subset
Test whether A and B are in the same subset
If A and B are in the same subset, then do nothing, else unify the subsets to which A and B belong

Union and Find

The Union-Find problem starts with n elements (numbered 1 to n), each one representing a singleton set. We allow two operations to be performed:
Post: #2
can explain deep about probability and stochiastics
Post: #3

In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not.
This definition of disjoint sets can be extended to any family of sets. A family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint. For example, the collection of sets { {1}, {2}, {3}, ... } is pairwise disjoint.

Two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two infinite sets whose intersection is a finite set may be said to be almost disjoint.

In topology, there are various notions of separated sets with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods. Similarly, in a metric space, positively separated sets are sets separated by a nonzero distance.

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