Can two sets be partially disjoint?

Yes, two sets can have some elements in common and still be partially disjoint. However, to be classified as disjoint, two sets must have no elements in common at all.

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Disjoint Set

Two sets are considered disjoint if they do not share any common elements. For a collection of two or more sets to be classified as disjoint, the intersection of all the sets within that collection must be empty.

1.0What is a Disjoint Set?

In mathematics, a disjoint set refers to a collection of sets that do not share any elements. This means that the intersection of any two disjoint sets is empty. If A and B are two sets, they are said to be disjoint if:

$A \cap B = ϕ$

Disjoint sets A and B can be shown by Venn diagram as follows-

2.0Examples of Disjoint Sets

Example 1: Consider the sets:

A = {1, 2, 3}

B = {4, 7, 8}

Here, A and B are disjoint sets because they have no elements in common.

Example 2: For the sets:

C = {x, y}

D = {z}

C and D are also disjoint since C \cap D=\phi .

3.0Properties of Disjoint Sets

Empty Intersection: The defining property of disjoint sets is that their intersection is always empty.
Cardinal number of Union of Disjoint Sets: If A and B are disjoint, then the cardinal number of union of these sets can be expressed as: $A \cup B = ∣ A ∣ + ∣ B ∣$

Where |A| and |B| are the cardinalities (number of elements) of the sets A and B respectively, while $∣ A \cup B ∣$ denotes cardinal number of $∣ A \cup B ∣$ .

More than Two Sets: A collection of sets can be disjoint if every pair of sets in the collection has an empty intersection. For example, sets A, B, C are disjoint if: $A \cap B = ϕ and A \cap C = ϕ and B \cap C = ϕ$

4.0Applications of Disjoint Sets

Venn Diagrams: In set theory, disjoint sets are often represented in Venn diagrams where circles representing sets do not overlap.
Probability: In probability theory, disjoint events are those that cannot occur simultaneously. If A and B are disjoint events, then: $P (A \cup B) = P (A) + P (B)$

5.0Solved Example on Disjoint Sets

Example 1: Show that Set A = {1, 3, 5} and Set B = {2, 4, 6} are Disjoint Sets.

Solution:

Given:

Set A = {1, 3, 5}

Set B = {2, 4, 6}

To prove: Sets A and B are disjoint sets.

Proof: Two sets are considered disjoint if their intersection results in the null set.

Therefore,

$A \cap B = {1, 3, 5} \cap {2, 4, 6}$

As we can see, sets A and B do not have any common elements.

So, $A \cap B = ϕ$

Hence, we conclude that A and B are disjoint sets.

Example 2: Are Set P = {5, 10, 15} and Set Q = {15, 20, 25} Disjoint Sets? If No, Justify Your Answer.

Solution:

Given:

Set P = {5, 10, 15}

Set Q = {15, 20, 25}

To check: Are P and Q disjoint?

Calculation:

$P \cap Q = {5, 10, 15} \cap {15, 20, 25}$ ={15}

Since the intersection of sets P and Q contains a common element {15}, it implies that P and Q are not disjoint sets.

Example 3: State whether Set A = {x, y, z} and Set B = {a, b, c} are Disjoint Sets or Not.

Solution:

Given:

Set A = {x, y, z}

Set B = {a, b, c}

To find: Are A and B disjoint?

Calculation:

$A \cap B = {x, y, z} \cap {a, b, c}$

Since there are no common elements between sets A and B:

$A \cap B = ϕ$

Therefore, sets A and B are disjoint sets.

Example 4: Are Set C = {2, 4, 6} and Set D = {6, 8, 10} Disjoint Sets? If No, Justify Your Answer.

Solution:

Given:

Set C = {2, 4, 6}

Set D = {6, 8, 10}

To check: Are C and D disjoint?

Calculation:

$C \cap D = {2, 4, 6} \cap {6, 8, 10}$ = {6}

Since the intersection of the two sets C and D results in a common element {6}, therefore, C and D are not disjoint sets.

Example 5: State whether Set E = {1, 2, 3, 4} and Set F = {5, 6, 7} are Disjoint Sets.

Solution:

Given:

Set E = {1, 2, 3, 4}

Set F = {5, 6, 7}

To find: Are E and F disjoint?

Calculation:

$E \cap F = {1, 2, 3, 4} \cap {5, 6, 7}$

As there are no common elements: $E \cap F = ϕ$

Thus, sets E and F are disjoint sets.

6.0Practice Questions on Disjoint Sets

Let A = {1, 3, 5, 7} and B = {2, 4, 6, 8}. Are sets A and B disjoint? If yes, justify your answer by finding their intersection.
If C = {10, 15, 20, 25} and D = {5, 10, 15}, are C and D disjoint sets? Support your answer with appropriate calculations.
Consider two sets E = {a, b, c, d} and F = {e, f, g, h}. Are E and F disjoint sets? If yes, write down their intersection.
Let G = {1, 2, 3, 4, 5} and H = {6, 7, 8, 9}. Find the union of sets G and H. Are they disjoint?
Determine if the following sets are disjoint or not:

A = {p, q, r, s} and B = {r, s, t, u}

If they are not disjoint, find their intersection.

Given the sets:

A = {10, 20, 30}

B = {40, 50, 60}

C = {70, 80, 90}

Verify if A, B, and C are pairwise disjoint. In other words, check if $A \cap B = ϕ$ , $A \cap$ $C = ϕ$ , and $B \cap C = ϕ .$

7.0Sample Questions on Disjoint Sets

What is a disjoint set?

Ans: A disjoint set is a set that has no elements in common with another set. In other words, the intersection of two disjoint sets is the empty set $(A \cap B = ϕ)$ .

How can you determine if two sets are disjoint?

Ans: To determine if two sets A and B are disjoint, you need to find their intersection. If $A \cap B = ϕ$ , then the sets are disjoint. If they share any elements, they are not disjoint.

Can more than two sets be disjoint?

Ans: Yes, a collection of more than two sets can be disjoint if every pair of sets within that collection has no elements in common. For instance, sets A, B, and C are all disjoint if

$A \cap B = ϕ, A \cap C = ϕ, and B \cap C = ϕ$ .

What is the union of disjoint sets?

Ans: The union of disjoint sets combines all elements from each set without duplication. For disjoint sets A and B, the union is given by:

$A \cup B = ∣ A ∣ + ∣ B ∣$ where |A| and |B| are the cardinalities (number of elements) of the sets.

Join ALLEN!

(Session 2026 - 27)

Name

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Class

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Preferred Mode

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I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Disjoint Set

Two sets are considered disjoint if they do not share any common elements. For a collection of two or more sets to be classified as disjoint, the intersection of all the sets within that collection must be empty.

1.0What is a Disjoint Set?

In mathematics, a disjoint set refers to a collection of sets that do not share any elements. This means that the intersection of any two disjoint sets is empty. If A and B are two sets, they are said to be disjoint if:

$A \cap B = ϕ$

Disjoint sets A and B can be shown by Venn diagram as follows-

2.0Examples of Disjoint Sets

Example 1: Consider the sets:

A = {1, 2, 3}

B = {4, 7, 8}

Here, A and B are disjoint sets because they have no elements in common.

Example 2: For the sets:

C = {x, y}

D = {z}

C and D are also disjoint since C \cap D=\phi .

3.0Properties of Disjoint Sets

Empty Intersection: The defining property of disjoint sets is that their intersection is always empty.
Cardinal number of Union of Disjoint Sets: If A and B are disjoint, then the cardinal number of union of these sets can be expressed as: $A \cup B = ∣ A ∣ + ∣ B ∣$

Where |A| and |B| are the cardinalities (number of elements) of the sets A and B respectively, while $∣ A \cup B ∣$ denotes cardinal number of $∣ A \cup B ∣$ .

More than Two Sets: A collection of sets can be disjoint if every pair of sets in the collection has an empty intersection. For example, sets A, B, C are disjoint if: $A \cap B = ϕ and A \cap C = ϕ and B \cap C = ϕ$

4.0Applications of Disjoint Sets

Venn Diagrams: In set theory, disjoint sets are often represented in Venn diagrams where circles representing sets do not overlap.
Probability: In probability theory, disjoint events are those that cannot occur simultaneously. If A and B are disjoint events, then: $P (A \cup B) = P (A) + P (B)$

5.0Solved Example on Disjoint Sets

Example 1: Show that Set A = {1, 3, 5} and Set B = {2, 4, 6} are Disjoint Sets.

Solution:

Given:

Set A = {1, 3, 5}

Set B = {2, 4, 6}

To prove: Sets A and B are disjoint sets.

Proof: Two sets are considered disjoint if their intersection results in the null set.

Therefore,

$A \cap B = {1, 3, 5} \cap {2, 4, 6}$

As we can see, sets A and B do not have any common elements.

So, $A \cap B = ϕ$

Hence, we conclude that A and B are disjoint sets.

Example 2: Are Set P = {5, 10, 15} and Set Q = {15, 20, 25} Disjoint Sets? If No, Justify Your Answer.

Solution:

Given:

Set P = {5, 10, 15}

Set Q = {15, 20, 25}

To check: Are P and Q disjoint?

Calculation:

$P \cap Q = {5, 10, 15} \cap {15, 20, 25}$ ={15}

Since the intersection of sets P and Q contains a common element {15}, it implies that P and Q are not disjoint sets.

Example 3: State whether Set A = {x, y, z} and Set B = {a, b, c} are Disjoint Sets or Not.

Solution:

Given:

Set A = {x, y, z}

Set B = {a, b, c}

To find: Are A and B disjoint?

Calculation:

$A \cap B = {x, y, z} \cap {a, b, c}$

Since there are no common elements between sets A and B:

$A \cap B = ϕ$

Therefore, sets A and B are disjoint sets.

Example 4: Are Set C = {2, 4, 6} and Set D = {6, 8, 10} Disjoint Sets? If No, Justify Your Answer.

Solution:

Given:

Set C = {2, 4, 6}

Set D = {6, 8, 10}

To check: Are C and D disjoint?

Calculation:

$C \cap D = {2, 4, 6} \cap {6, 8, 10}$ = {6}

Since the intersection of the two sets C and D results in a common element {6}, therefore, C and D are not disjoint sets.

Example 5: State whether Set E = {1, 2, 3, 4} and Set F = {5, 6, 7} are Disjoint Sets.

Solution:

Given:

Set E = {1, 2, 3, 4}

Set F = {5, 6, 7}

To find: Are E and F disjoint?

Calculation:

$E \cap F = {1, 2, 3, 4} \cap {5, 6, 7}$

As there are no common elements: $E \cap F = ϕ$

Thus, sets E and F are disjoint sets.

6.0Practice Questions on Disjoint Sets

Let A = {1, 3, 5, 7} and B = {2, 4, 6, 8}. Are sets A and B disjoint? If yes, justify your answer by finding their intersection.
If C = {10, 15, 20, 25} and D = {5, 10, 15}, are C and D disjoint sets? Support your answer with appropriate calculations.
Consider two sets E = {a, b, c, d} and F = {e, f, g, h}. Are E and F disjoint sets? If yes, write down their intersection.
Let G = {1, 2, 3, 4, 5} and H = {6, 7, 8, 9}. Find the union of sets G and H. Are they disjoint?
Determine if the following sets are disjoint or not:

A = {p, q, r, s} and B = {r, s, t, u}

If they are not disjoint, find their intersection.

Given the sets:

A = {10, 20, 30}

B = {40, 50, 60}

C = {70, 80, 90}

Verify if A, B, and C are pairwise disjoint. In other words, check if $A \cap B = ϕ$ , $A \cap$ $C = ϕ$ , and $B \cap C = ϕ .$

7.0Sample Questions on Disjoint Sets

What is a disjoint set?

Ans: A disjoint set is a set that has no elements in common with another set. In other words, the intersection of two disjoint sets is the empty set $(A \cap B = ϕ)$ .

How can you determine if two sets are disjoint?

Ans: To determine if two sets A and B are disjoint, you need to find their intersection. If $A \cap B = ϕ$ , then the sets are disjoint. If they share any elements, they are not disjoint.

Can more than two sets be disjoint?

Ans: Yes, a collection of more than two sets can be disjoint if every pair of sets within that collection has no elements in common. For instance, sets A, B, and C are all disjoint if

$A \cap B = ϕ, A \cap C = ϕ, and B \cap C = ϕ$ .

What is the union of disjoint sets?

Ans: The union of disjoint sets combines all elements from each set without duplication. For disjoint sets A and B, the union is given by:

$A \cup B = ∣ A ∣ + ∣ B ∣$ where |A| and |B| are the cardinalities (number of elements) of the sets.

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