Disjoint Set

Introduction

Disjoint sets can be used for a number of math problems but specifically in data structures. In set theory, disjoint sets are two sets which do not share any common observations. In other words, if we take the intersection of two sets and the resulting set is an empty set then the sets are said to be disjoint.. In this tutorial, we will learn what are sets, disjoint sets, and conditions for sets to be disjoint along with some of the solved examples.

Sets

A set is a collection of elements or observations for mathematical modelling. Elements of a set can be any mathematical object such as numbers, variables, points in space, etc.

Disjoint Sets

For two sets to be disjoint, their intersection should be an empty set or null. For more than two sets, when the intersection between all the sets is null or an empty set.

A disjoint set has no common element between them. For example, if there are two sets, A = {x, y} and B = {p, q}. As we can see that there are no common elements between the sets. We can take the intersection of two sets.

$$\mathrm{A ∩ B = 𝛟.}$$

Therefore, the intersection of two disjoint sets results in a null set. Two sets X and Y can be represented with the help of Venn diagrams. The figure below represents two sets X and Y in the form of Venn diagrams.

Condition of Sets to be Disjoint

In order to determine whether two sets are disjoint, we have to find the intersection of the sets. Therefore, the condition for two sets to be disjoint can be given as −

$$\mathrm{A ∩ B = 𝛟.}$$

If there are more than two sets, intersection between each pair is calculated. Then the result is evaluated for any result that is not an empty or null set. If all the results are null then the group of sets are called disjoint sets.

For example, let there be three sets such as A= {4, 5}, B= {5, 3} and C= {1, 3}.

We have to determine the intersection between each pair of sets.

First pair

$$\mathrm{A∩B=\{4,5\} ∩\{5,3\}.}$$

$$\mathrm{A∩B=\{5\}}$$

Second pair

$$\mathrm{B∩C=\{5,3\} ∩\{1,3\}}$$

$$\mathrm{B∩C=\{3\}}$$

Third pair

$$\mathrm{C∩A=\{1,3\} ∩\{4,5\}}$$

$$\mathrm{C∩A=Φ}$$

Therefore, we can say that out of the three pairs A∩B, B∩C, and C∩A, Pair C and A are disjoint sets.

Mutually disjoint sets

Mutually disjoint sets are sets which are a subset of the same set but are disjoint sets. For example, let there be a set X, A and B are subsets of X such that A ≠ B,and A ∩ B = ϕ. Therefore, A and B are called mutually disjoint sets.

Solved Examples

1)If there are two sets, P = {11, 13} and Q = {1, 7}, find if the sets are disjoint sets or not.

Answer:

Given that: P = {11, 13} and Q = {1, 7}

First, we see if the sets are disjoint. Therefore the intersection of the given sets is calculated.

$$\mathrm{P∩Q=Φ}$$

As we know that if the intersection between two sets is null then the sets are known as disjoint sets. Therefore, P and Q are considered disjoint sets.

2)Determine whether A and B are disjoint sets. A = {a, i, o, u} and B= {f, h,w,o}.

Answer:

Given: A = {a, i, o, u}, B= {f, h, w, o}

First, we see if the sets are disjoint. Therefore, the intersection of the given sets is calculated.

$$\mathrm{A∩B=\{o\}}$$

Now we know that for two sets to be disjoint the intersection between them should be an empty set.

Therefore, the given sets, A and B, are not disjoint sets.

3)Prove that the given sets A, B, and C are disjoint sets or not. A = {3, 4, 7}, Y = {1, 7, 9} and Z = {7, 2}

Answer: To prove that all three given sets are disjoint sets, we have to prove that each pair are disjoint sets. We have to find A∩B, B∩C, and C∩A

$$\mathrm{A∩B=\{3,4,7\}∩\{1,7,9\}}$$

$$\mathrm{A∩B=\{7\}}$$

$$\mathrm{B∩C=\{1,7,9\}∩\{7,2\}}$$

$$\mathrm{B∩C=\{7\}}$$

$$\mathrm{C∩A=\{7,2\}∩\{3,4,7\}}$$

$$\mathrm{C∩A=\{7\}}$$

From the results above, we can see that none of the intersections between the pair of sets is an empty set. Therefore, we can say that the given sets A, B, and C are not disjoint sets.

4)Find whether the sets X = {weekdays} and Y = {weekends} are disjoint sets.

Answer: Given that

Set X= {Monday, Tuesday, Wednesday, Thursday, Friday}

Set Y= {Saturday, Sunday}

We know that for two sets to be disjoint the intersection between them should be an empty set. Finding the intersection between the two sets.

X∩Y= {Monday, Tuesday, Wednesday, Thursday, Friday} ∩ {Saturday, Sunday}

X∩Y= ɸ

Therefore, we can say that the given sets X and Y are disjoint sets.

5)Determine whether A and B are disjoint sets, A = {2, 4, 6, 8, 10} (Set of first 5 positive even integers) and B= {2, 3, 5, 7, 11} (Set of first 5 primes).

Answer:

Given: A = {2, 4, 6, 8, 10}, B= {2, 3, 5, 7, 11}

First, we see if the sets are disjoint. Therefore, the intersection of the given sets is calculated.

$$\mathrm{A ∩ B = \{2\}}$$

Now we know that for two sets to be disjoint the intersection between them should be an empty set.

Therefore, the given sets, A and B, are not disjoint sets.

Conclusion

In set theory, disjoint sets are two sets which do not have any common element. A set is a collection of elements or observations for mathematical modelling. A disjoint set has no common element between them.

For example, if there are two sets, A = {x, y} and B = {p, q}. The intersection of two disjoint sets results in a null set. If there are more than two sets, intersection between each pair is calculated. Mutually disjoint sets are sets which are a subset of the same set but are disjoint sets. Let there be a set X, A and B are subsets of X such that A ≠ B, and A ∩ B = ϕ. Therefore, A and B are called mutually disjoint sets.

FAQs

1. Define the union between two disjoint sets?

A union between disjoint sets is a binary operation done on any two disjoint sets. A disjoint union is a bijective operation.

2. Define the condition for two sets to be disjoint?

Two sets P and Q can be called disjoint sets if the intersection between them results in a null set. A ∩ B = 𝛟.

3. What are pairwise disjoint sets?

Two sets can be called pairwise disjoint sets if for a set X, A and B are subsets of X such that A ≠ B, and A ∩ B = ϕ.

4. How can two disjoint sets be expressed with venn-diagrams?

Two disjoint sets can be expressed with the help of two circles with no common elements in the form of venn-diagrams.