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Disjoint Sets

In set theory, sometimes we find that there is no common elements in two sets or we can say that the intersection of the sets are an empty set. This type of sets are called the disjoint sets. If we have A = {1,2,3} and B = {4,5,6}, then we can say that the given two sets are disjoint, since there are no common elements in these two.

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Disjoint Sets Definition

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

For example, {p,q,r} and {b,a,k} are disjoint sets. Formally, two sets A and B are disjoint sets if the intersection of them are the empty set i.e. if A $\cap$ B = $\phi$.

Pairwise Disjoint Sets

We can extend the definition of disjoint set to any collection of sets. A collection of sets is pairwise disjoint or mutually disjoint, if any two sets in the collection are disjoint.

Let S be the set of any collection of sets and X and Y are two sets in S i.e. X, Y $\in$ S. Then, S is known as pairwise disjoint if and only if
X $\neq$ Y. Therefore, X $\cap$ Y = $\phi$

Examples:
1. S = { {a}, {b,d}, {e,f,g} } is pairwise disjoint set.
2. S = { {b,c}, {c,d} } is not pairwise disjoint set, since we have c as the common element in two sets.

Disjoint Set Union

This is a binary operation on two sets. The elements of any disjoint union can be expressed in terms of ordered pair as (x, j), where j is the index that indicates that set where the element x came from. With the help of this operation, we can combine all the different(distinct) elements of a pair of sets.

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

The disjoint union of sets A = ( a, b, c, d ) and B = ( e, f, g, h ) is as follows:
A* = { (a,0), (b,0), (c,0), (d, 0) } and B* = { (e,1), (f,1), (g,1), (h,1) }
Then,
A U* B = A* U B*
= { (a,0), (b,0), (c,0), (d, 0), (e,1), (f,1), (g,1), (h,1) }

Examples of Disjoint Sets

Given below are some of the examples on disjoint sets.

Solved Examples

Question 1: Prove that the following two sets are disjoint sets.

G = {p, q, r, s}

H = {x, y}

Solution:

The intersection of set H and set G gives an empty set. Here, set G and H does not have the elements in common with each other.

That is, G $\cap$ H = { }

Hence, the sets G and H are disjoint sets.

Question 2: Prove that Set G = {10, 12, 20, 18, 25} and set H = {11, 17, 27, 44} are disjoint sets.
Solution:

In the above problem, we have no common elements in G and H.

These elements are not intersecting of two elements.

G $\cap$ H = { }

Hence, the two sets G and H are disjoint sets.

Question 3:

Draw a Venn diagram to represent the disjoints between the sets

W = {22,14,55,77,99,101,200} and Z = {21,23,57,9,75,103}

Solution:

In the given problem, we have no common factor. So, the given sets are disjoint.

We find that W $\cap$ Z = { } in both W and Z are Disjoint

Given below is the Venn diagram,

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

Question 4:

By using Venn Diagram, prove that the following sets are not disjoint

W = {22,14,90,42,99} and Z = {77,15,57,9}

Solution:

We find that W $\cup$ Z = {22,14} in both W and Z are not Disjoint.

Hence, we find common factors

Given below is the Venn diagram,

The above Venn diagram clearly shows that the given sets are not disjoint sets.

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