What is Disjoint Set: Definition and Examples

Disjoint sets are two sets A₁ and A₂ if their intersection A₁ ∩ A₂ ≡ Φ, where Φ is the empty set. In this case for example, {A, B, C} and {D, E} are termed as disjoint sets, but on the contrary {A, B, C} and {C, D, E} are not disjoint sets. Mutually exclusive or independent sets are sometimes known as disjoint sets. Two disjoint sets have the same intersection, which is the empty set.

Read More: Bayes Theorem Formula

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Disjoint Set

[Check on Sample Questions]

Consider two sets A and B to better understand the Venn diagram representation of disjoint sets and let A= {2, 4, 6, 8} , B = {3, 9, 12, 10}. 

Because the two sets, A and B, do not have any crossing elements, they are disjoint sets with no overlapping regions in their Venn diagrams.

Therefore we can say that Disjoint sets are those that don't share any elements. Their intersection always yields a null or empty set.

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Process to Determine if Two Sets are Disjoint

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The intersection operation is all that is required to establish whether two sets are disjoint. There will never be a shared element between disjoint sets. As a result, the intersection of their sets will always be null. Hence the null set will be a disjoint set.

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Now taking an example we can understand the concept of disjoint set:

Example: Take two sets, A= {1, 2, 3} and B= {a, b, c} now through the given values construct a venn diagram and point out their relationship based on it.

Solution: Analyzing the two sets reveals that they do not contain any intersecting elements, implying that A ∩ B = Φ, Hence we can say that the set A and B are Disjoint sets.

Venn Diagram

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Pairwise Disjoint Set

[Check on Sample Questions]

The pair of sets whose intersection is a null set is known as pairwise disjoint sets. There is no common element between these sets. As a result, pairwise disjoint sets and disjoint sets are the same thing.

Hence the criteria or condition for a pairwise disjoint set are as follows:

A ∩ B = Φ

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In order to equip with concept of pairwise disjoint set a following example is provided below:

Example- Take two sets, A= {1, 2, 3} and B= {5, 6, 7} Find out whether the given sets are pairwise disjoint sets not?

Solution- The basic condition for a pairwise disjoint set is : A ∩ B = Φ

Now making intersection operation on sets A and B we get;

A ∩ B = {1, 2, 3} ∩ {5, 6, 7}

A ∩ B = Φ

Therefore, the two sets A and B are pairwise disjoint sets as there is no common element in both the sets.

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Disjoint Sets Union

A binary operation on two sets is a disjoint set union. Any disjoint union's elements can be defined in terms of ordered pairs as (a, b), where b is the index of the member a's origin. We can unite all the different (distinct) components of a pair of sets using this method.

A*B = (A0(B1) A*B*

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With the help of an example the disjoint union can be understood:

Example- Given a set, A= {6, 7, 8, 9, 10} and B= {6, 7, 8, 9} determine the disjoint of union sets?

A*= {(6,0), (7,0), (8,0), (9,0), (10,0)} and B*= {(6,1), (7,1), (8,1), (9,1)},

Solution- Solving, A*B = A*B*

= {(6,0), (7,0), (8,0), (9,0), (10,0), (6,1), (7,1), (8,1), (9,1)}

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Points to Remember

• Disjoint sets is a part of unit 1 Sets and function, chapter 1 Sets and carries a total of 6 to 8 marks.
• Disjoint sets always have unique elements.
• If there is a common element in any given set then it is not a disjoint term.
• Disjoint set union is determined using the below formula: A*B = (A0(B1) A*B*
• The pair of sets whose intersection is a null set is known as pairwise disjoint sets.

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