Disjoint Sets

Two sets are said to be disjoint if there are no common elements, In other words, when the intersection of the two sets is empty, then those sets are said to be disjoint sets. Disjoint sets have applications in data structures and are used in solving a number of math problems.

Let us learn more about disjoint set definition, condition for disjoint sets, their venn diagram, pairwise disjoint sets, and union of disjoint sets, with the help of examples, FAQs.

There are no common elements in disjoint sets because the intersection set operation between them will always result in a null set or an empty set. Let's consider two distinct sets X = {a, b} and Y = {c, d}. It is evident that these two sets do not have any common elements between them.

• The intersection of these two sets X and Y can be written as: X ∩ Y = 𝛟.
• Thus, we can say that the intersection operation on disjoint sets will yield a null set.

Disjoint Sets Definition

Two sets are referred to as disjoint sets if their intersection is a null or empty set. For a collection of two or more sets, the condition of disjointness holds if the intersection of the entire collection is an empty set.

We can use Venn diagram to portray the relationship between the given sets. Most of the set operations can be illustrated using Venn diagrams. Similarly, we can use Venn diagrams to represent disjoint sets too. The Venn diagram of disjoint sets has no overlapping region between them.

Let's consider two sets X = {2, 4, 6, 8} and let Y = {1, 3, 5, 7}. The Venn diagram of these two sets can be shown below:

Since the two given sets, X and Y, do not have any intersecting or common element, we can consider these two sets as disjoint sets, and thus the Venn diagram between two disjoint sets will not contain any overlapping region.

In order to find if two sets are disjoint sets, we need to perform the intersection of sets operation on these two sets. The condition for any given sets to be disjoint can be given as A ∩ B = 𝛟

• If you have more than two sets, then you can check if they are disjoint sets by taking intersections between pairs of sets.
• Once the intersection is done, we need to evaluate each pair of sets’ result and if any of these pairs of sets results in an empty set, then that pair of sets can be considered as disjoint sets.

Let's consider an example. P = {1, 2}, Q = {2, 3} and R = {5, 3}. Let's determine if any of these sets are disjoint sets.

The first step is to perform the intersection operation on each pair of sets.

P ∩ Q = {1, 2} ∩ {2, 3}

P ∩ Q = {2}

And,

Q ∩ R = {2, 3} ∩ {5, 3}
Q ∩ R = {3}

And then the last pair,

P ∩ R = {1, 2} ∩ {5, 3}

P ∩ R = 𝛟

So, among the given sets, P and R can be considered as disjoint sets.

The term pairwise disjoint refers to a family of collections of subsets. Let A be the set of a group of sets where P and Q are two sets in set A. Then, P and Q are called pairwise disjoint sets if and only if P and Q are subsets of A, P ≠ Q, and P ∩ Q = ϕ. These are also called mutually disjoint sets.

If we consider a set of sets, then whichever pair of sets in that group results in a null intersection, then that pair of sets can be considered as pairwise disjoint sets. Any two sets can be considered as pairwise disjoint sets when their intersection results in a null set. These sets do not contain any common elements between them. Thus, we can consider pairwise disjoint sets similar to disjoint sets. The criteria for any two sets to be pairwise disjoint sets is given as:

P ∩ Q = 𝛟, P, Q ∈ A, P ≠ Q

We know that when we perform union operation on two sets, then the resultant set will contain all the elements that are present individually in each of the given sets. The union of sets that are disjoint is a little different from the usual union of sets.

• The disjoint union can be considered as a binary operation that is performed on any two disjoint sets.
• The first thing is to make sure that the sets follow the disjoint status even after the union operation is performed.
• For this reason, it is essential to alter each of the disjoint sets before performing a disjoint union on them.
• Thus, the disjoint of union needs to follow this set notation: P U* Q = { P x (0) } U { Q x (1) }, Where U* represents the disjoint union and sets P and Q are disjoint sets.
• The disjoint union can be considered as a bijective operation.

The convention that is mentioned above helps to obtain the disjoint sets’ union and also makes sure that the disjoint sets always retain their disjoint identity. Let's consider an example to understand this concept. Take the two disjoint sets: P = {p, q, r} and Q = {s, t, v}. Let's find the union of these sets.

Since sets P and B are disjoint sets, their union will result in a disjoint union operation. So, the disjoint union of P and Q is:

P U* Q = {(p,0), (q,0), (r,0)} U {(s, 1), (t, 1), (v, 1)}

So,

P U* Q = {(p,0), (q,0), (r,0)},{(s, 1), (t, 1), (v, 1)}

Where P U* Q is the disjoint union of the disjoint sets P and Q.

Related Articles on Disjoint Sets

Check out the following pages related to disjoint sets

• Finite and Infinite Sets
• Sets Formula
• Intersection of Sets

Important Notes on Disjoint Sets

Here is a list of a few points that should be remembered while studying disjoint sets

• There are no common elements in disjoint sets because the intersection set operation between them will always result in a null set or an empty set.
• Any two sets can be considered as pairwise disjoint sets when their intersection results in a null set.

What Are Disjoint Sets in Math?

Disjoint sets are sets that have no common elements. Their intersection will always result in a null set or an empty set. Let's consider two distinct sets X = {a, b} and Y = {c, d}. It is evident that these two sets do not have any common elements between them.

• The intersection of these two sets X and Y can be written as: X ∩ Y = 𝛟.
• Thus, we can say that the intersection operation on disjoint sets will yield a null set.

How Do You Find the Disjoint Set?

In order to find if two sets are disjoint sets, we need to perform the intersection of set operation on them. The condition for the given sets to be disjoint can be given as A ∩ B = 𝛟

• If you have more than two sets, then you can check if they are disjoint sets by taking intersections between pairs of sets.
• On evaluating the intersection of each pair of sets, if any of these pairs of sets result in an empty set, then that pair of sets can be considered as disjoint sets.

Can Two Null Sets Be Disjoint?

Yes. Since the intersection of any set with a null set will result in a null set, the intersection of two null sets will also be a null set and hence they can be considered as disjoint sets.

What Is the Condition for Two Sets To Be Disjoint Sets?

For two sets, say A and B, can be considered as disjoint sets, if their intersection is the null set, that is A ∩ B = 𝛟.

Can Two Disjoint Sets Have a Union?

Yes, two disjoint sets can have a union. The disjoint union can be considered as a bijective set operation. The disjoint of union needs to follow this convention: P U* Q = { P x (0) } U { Q x (1) }, Where U* represents the disjoint union and sets P and Q are disjoint sets.

What Is the Union of Two Disjoint Sets?

The disjoint union can be considered as a binary operation that is performed on any two disjoint sets. The disjoint union can be considered as a bijective operation. Given below are the steps for the union of disjoint sets:

• The sets should follow the disjoint status even after the union operation is performed.
• Thus, the disjoint of union follows the convention: P U* Q = { P x (0) } U { Q x (1) }, U* represents the disjoint union and sets P and Q are disjoint sets.

What Are Mutually Disjoint Sets?

We can consider two sets, say X and Y as mutually disjoint sets if

• X, Y ∈ A
• X ≠ Y
• X ∩ Y =∅.