Disjoint Set

In Set theory, sometimes we notice that there are no common elements in two sets or we can state that the intersection of the sets is an empty set or null set. This type of set is called a disjoint set. If we have X = {a, b, c} and Y = {d, e, f}, then we can say that the given two sets are disjoint, since there are no common elements in these two sets X and Y.

Disjoint Set Definition

Two sets are said to be disjoint when they have no common element.

Consider an example, {1, 2, 3} and {4, 5, 6} are disjoint sets.

Two sets A and B are disjoint sets if the intersection of two sets is a null set or an empty set.

i.e. A ∩ B = ϕ

Pairwise Disjoint sets

We can proceed with the definition of a disjoint set to any group of sets. A collection of sets is pairwise disjoint if any two sets in the collection are disjoint. It is also known as mutually disjoint sets.

Let P be the set of any collection of sets and A and B are two sets in P

i.e. A, B ∈ P. Then, P is known as pairwise disjoint if and only if A ≠ B. Therefore, A ∩ B = ϕ

Examples:

  • P = { {1}, {2, 3}, {4, 5, 6} } is a pairwise disjoint set.
  • P = { {1, 2}, {2, 3} } is not pairwise disjoint set, since we have 2 as the common element in two sets

Disjoint Set Union

A disjoint set union is a binary operation on two sets. The elements of any disjoint union can be described in terms of ordered pair as (x, j), where j is the index, that represents the origin of the element x. With the help of this operation, we can join all the different (distinct) elements of a pair of sets.

The disjoint union is denoted as X U* Y = ( X x {0} ) U ( Y x {1} ) = X* U Y*

Assume that,

The disjoint union of sets X = ( a, b, c, d ) and Y = ( e, f, g, h ) is as follows:

X* = { (a,0), (b,0), (c,0), (d, 0) } and Y* = { (e,1), (f,1), (g,1), (h,1) }

Then,

X U* Y= X* U Y*

Therefore, the disjoint union set is { (a,0), (b,0), (c,0), (d, 0), (e,1), (f,1), (g,1), (h,1) }

Disjoint Sets Examples

Question 1:

Show that the given two sets are disjoint sets.

A = {5, 9, 12, 14}

B = {3, 6}

Solution:

Given: A = {5, 9, 12, 14}, B = {3, 6}

The intersection of set A and set B gives an empty set.

A ∩ B = {5, 9, 12, 14} ∩ {3, 6}

Here, set A and B do not have any common element

That is, A ∩ B = { }

Hence, the sets A and B are disjoint sets.

Question 2:

Draw a Venn diagram to represent the disjoints between the sets

X = {a, b, c, d, f} and Y = {e, g, h, i}

Solution:

Given: X = {a, b, c, d, f}, Y = {e, g, h, i}

In the given problem, we don’t have a common factor.

Therefore, the given sets are disjoint.

We find that A ∩ B = { }

The Venn diagram for the given disjoint set is

Disjoint Set - Venn Diagram

The Venn diagram clearly shows that the given sets are disjoint sets.

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