Disjoint sets

In set theory, sometimes we discover that there is no common element in two sets or we can say that the intersection of the sets is an empty set. These types of sets are known as 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.

Disjoint Sets Definition

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

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.

Pairwise Disjoint Sets

We can extend the definition of disjoint set to any group of sets. A group of sets is pairwise disjoint or mutually disjoint, if any two sets in the group 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 referred to as pairwise disjoint if and only if

X ≠ Y. Therefore, X ∩ Y = Φ

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 articulated in terms of structured pair as (x, j), where j is the index that indicates the set from where the element x came from. With the assistance of this operation, we can join 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

Provided below are a few 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 is an empty set. Here, set G and H does not have the elements in common with each other.

That is, G ∩ H = { }

Therefore, 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 question, we have no common elements in G and H.

These elements are not intersecting of two elements.

G ∩ H = { }

therefore, 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 question, we have no common element. Therefore, the given sets are disjoint.

We find that W ∩ Z = { } in both W and Z are Disjoint

The above Venn diagram vividly illustrates that the given sets are disjoint sets.

Question 4:

By making use of Venn diagram, prove that the following sets are not disjoint

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

Answer:

We discover that W ∩ Z = {22, 14} in both W and Z are not Disjoint.

Therefore, we find common factors

The above Venn diagram vividly shows that the given sets are not disjoint sets. View more on psalmfresh.blogspot.com for more educating and tricks article Thank you