Disjoint Set

A set is a mathematical model for a collection of diverse things; it contains elements or members, which can be any mathematical object: numbers, symbols, points in space, lines, other geometrical structures, variables, or even other sets. The empty set is a set with no elements, while a singleton is a set with only one element. A set can either have a finite or infinite number of elements.

What do you name a set who don’t have anything in common? What about the sets that don’t share any common elements? It will be referred to as a disjoint set-in mathematical terms if the intersection of the set is fully empty or null. With the help of an example, we can better comprehend this. Assume you have two sets, one with the numbers 1,2,3,4,5 and the other with the numbers 6,7,8,9,10.

Is there anything you can see in common between the two sets? You can’t since there are no elements in common between them. As a result, we can refer to it as a disjoint set. Let’s take a walk together and talk about a few more key points that we need to grasp in order to fully grasp the notion we just discussed. We’ll learn more about disjoint sets and disjoint set unions as we progress. Not only that, but we’d look at a few examples to make sure we understood what a disjoint set was.

Definition of a Disjoint Set

Disjoint set meaning can be defined as two sets that share no common element. If a collection contains two or more sets, the intersection of the entire collection must be empty, according to the disjoint set definition.

A group of sets could have a null intersection without being a disjoint set, for example. Consider the following three sets: {{11, 12}, { 12, 13}, {11, 13}}. They do have a null intersection, but we can’t call it a disjoint set because the intersection of a group of one set is identical to that set, which can be regarded a non-empty set even though there are no pairs to compare.

Consider two pairs A={1, 2, 3} and B={4, 5, 6} as another basic example. They’re the epitome of a disjointed set. In more precise terms, the intersection of sets A and B is empty, which is known as an empty set or a null set.

As a result, A ∩ B = ϕ

There is one distinction that we must understand in our minds: the difference between “the intersection of two sets” and “the difference of two sets.”

We’re only concerned with the junction in this case.

What is the difference between a joint and a disjointed set?

The distinction between a joint and a disjoint set has already been established. Let’s look at an example of a disjoint union to see if we’re right.

We’re going to use two sets, X and Y. Consider both the X and Y sets to be non-empty. As a result, X ⋂ Y holds true and can be referred to as non-empty sets or joint sets. But what should it be called if X ⋂ Y is an empty set? Yes, we’ll refer to it as a disjoint set.

Here’s an illustration:

X = {2, 5, 7} and Y = {1, 5, 6}

X ∩ Y = {5}

As a result, X and Y are both joint sets.

However, if

X = {1, 3, 7} and Y = {2, 5, 6}

X ∩ Y = Ø

As a result, X and Y are considered distinct sets.

=>Disjoint set union: 

A disjoint set union is a binary operation on any two sets. The element in a disjoint union can be stated using ordered pairs such as (x, j). The element x’s origin is represented by the letter j in this case. This procedure, in turn, joins all the distinct parts of a pair of sets together.

A disjoint necessitates the fulfilment of two requirements. The union of two or more disjoint sets is the first and most typical indication. Second, if they remain fragmented, the union of a disjunct set will be produced, with the sets being adjusted to obtain them before the altered sets’ union is formed.

When the elements of two sets are switched by an ordered pair of elements, the result is a disjoint set. A binary value also indicates whether the element belongs to the first or second collection. If the group comprises two or more sets, an ordered pair of the element and the set that contains it will be substituted for each element.

denotation of disjoint set => X U* Y = ( X x {0} ) U ( Y x {1} ) = X* U Y*

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

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*

As a result, the disjoint union set will be{ (a,0), (b,0), (c,0), (d, 0), (e,1), (f,1), (g,1), and (h,1)}, respectively.

Conclusion:

Two sets are said to be disjoint in ‘Set Theory’ if their intersection produces a null or empty set. These sets have no elements in common. Only one requirement exists for a set to be disjoint. The intersection of the two sets, in other words, should be an empty set.

For example, if you have two sets A = {1,2, 3} and B ={ 5, 6, 8}, they are disjoint since they share no elements, and their intersection is a null set.