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 ∩ B = ɸ

Thus, two empty sets are always disjoint sets.

If A = ɸ and B = ɸ are two sets, then ɸ ∩ ɸ = ɸ

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 ∩ B = {a, b, m, n} ∩ {3, 4, 9, 10} = { }

Thus, A and B are disjoint sets.

Pairwise Disjoint Sets

A group of sets X is called ‘pairwise disjoint’ if the intersection of each pair of sets in the group is an empty set.

If ‘X’ is a set containing various sets, where ‘A’ and ‘B’ are two sets in set X, then the condition for the sets A and B to be pairwise disjoint is defined as:

A ∩ B = ɸ, for A, B ∈ X and  A ≠ 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 ⨆ B = {A x (0)} U {B x (1)}

Here, 

• A and B are the disjoint sets
• The symbol ‘⨆’ 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 ⨆ B = {A x (0)} U {B x (1)}

⇒ A ⨆ B = {(a, 0), (b, 0), (c, 0)} U {(g, 1), (h, 1), (k, 1)}

⇒ A ⨆ B = {(a, 0), (b, 0), (c, 0), (g, 1), (h, 1), (k, 1)}

Joint Set vs. Disjoint Set

Joint Set	Disjoint Set
The sets share common elements.	The sets share no common elements.
The intersection of two joint sets is not an empty set.If A and B are two joint sets, then A ∩ B ≠ ɸ	The intersection of two disjoint sets is an empty set.If A and B are two disjoint sets, then A ∩ B = ɸ
For example,If A = {a, d, s} and B = {a, s, t}, then A ∩ B = {a, s} ≠ ɸThus, A and B are joint sets.	For example,If A = {1, 5, 14} and B = {3, 7, 16}, then A ∩ B = { } = ɸThus, A and B are disjoint sets.