# Disjoint Set

Disjoint sets are sets that share no common elements between them. Thus, the intersection between two disjoint sets is a null set.

For example, A = {a, b, m, n} and B = {3, 4, 9, 10} are disjoint, and A âˆ© B = É¸

Thus, two empty sets are always disjoint sets.

If A = É¸ and B = É¸ are two sets, then É¸ âˆ© É¸ = É¸

## Disjoint Sets Venn Diagram

When represented in a Venn diagram, disjoint sets show no overlapping regions. Drawing the Venn diagram of the two disjoint sets: A = {a, b, m, n} and B = {3, 4, 9, 10}, we get

## Verifying Disjoint Sets

If two sets are disjoint, their intersection will be a null set.

Considering the two sets A = {1, 2, 3, 4} and B = {5, 6, 7, 8}

A âˆ© B = {a, b, m, n} âˆ© {3, 4, 9, 10} = { }

Thus, A and B are disjoint sets.

Verify whether the sets A = {7, 9, 11} and B = {2, 3, 6} are disjoint sets.

Solution:

Here, A = {7, 9, 11} and B = {2, 3, 6}
The elements of set A are 7, 9, and 11
The elements of set B are 2, 3, and 6
A âˆ© B = {7, 9, 11} âˆ© {2, 3, 6} = { }, condition is satisfied.
Thus, A and B are disjoint sets.

Verify whether the sets A = {a, e, i} and B = {a, b, c} are disjoint sets.

Solution:

Here, A = {a, e, i} and B = {a, b, c}
The elements of set A are a, e, and i
The elements of set B are a, b, and c
A âˆ© B = {a, e, i} âˆ© {a, b, c} = {a}
The disjoint set condition is not satisfied.
Thus, A and B are not disjoint.

## Pairwise Disjoint Sets

A group of sets X is called â€˜pairwise disjointâ€™ if the intersection of each pair of sets in the group is an empty set.

If â€˜Xâ€™ is a set containing various sets, where â€˜Aâ€™ and â€˜Bâ€™ are two sets in set X, then the condition for the sets A and B to be pairwise disjoint is defined as:

A âˆ© B = É¸, for A, B âˆˆ X and  A â‰  B

These sets are also called mutually disjoint sets.

## Disjoint Set Union

When we combine two sets through the union operation, the resulting set includes all elements in either of the original sets. However, the concept of a disjointed union is slightly different from that of a regular union.

In disjoint unions, any two disjoint sets are combined using a binary operation. After executing the disjoint union operation, the resultant set must satisfy the disjoint union operation.

Mathematically, the disjoint union is represented as

A â¨† B = {A x (0)} U {B x (1)}

Here,

• A and B are the disjoint sets
• The symbol â€˜â¨†â€™ denotes the disjoint union and

This operation is thus bijective, ensuring that the disjoint sets retain their disjoint identity.

Now, let us consider two disjoint sets, A = {a, b, c} and B = {g, h, k}, and find their disjoint union.

Since A and B are disjoint sets, their union follows the disjoint union operation. Thus, the disjoint union of A and B is:

A â¨† B = {A x (0)} U {B x (1)}

â‡’ A â¨† B = {(a, 0), (b, 0), (c, 0)} U {(g, 1), (h, 1), (k, 1)}

â‡’ A â¨† B = {(a, 0), (b, 0), (c, 0), (g, 1), (h, 1), (k, 1)}

If A = {5, 8} and B = {11, 13} are two disjoint sets, then find the disjoint union of the given sets.

Solution:

Here, A = {5, 8} and B = {11, 13} are two disjoint sets.
As we know, A â¨† B = {A x (0)} U {B x (1)}
â‡’ A â¨† B = {(5, 0), (8, 0)} U {(11, 1), (13, 1)}
â‡’ A â¨† B = {(5, 0), (8, 0), (11, 1), (13, 1)}
Thus, the disjoint union of sets A and B is {(5, 0), (8, 0), (11, 1), (13, 1)}