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:

$A\cap B=\varphi$

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

2.0Examples of Disjoint Sets

1. 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.

2. 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

1. Empty Intersection: The defining property of disjoint sets is that their intersection is always empty.
2. 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 $|A\cup B|$.

3. 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:

$A\cap B=\varphi \text{ and }A\cap C=\varphi \text{ and }B\cap C=\varphi$

4.0Applications of Disjoint Sets

1. Venn Diagrams: In set theory, disjoint sets are often represented in Venn diagrams where circles representing sets do not overlap.
2. 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, $A\cap B=\varphi$

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:

$A\cap B=\varphi$

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: $E\cap F=\varphi$

Thus, sets E and F are disjoint sets.

6.0Practice Questions on Disjoint Sets

1. 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.
2. If C = {10, 15, 20, 25} and D = {5, 10, 15}, are C and D disjoint sets? Support your answer with appropriate calculations.
3. 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.
4. Let G = {1, 2, 3, 4, 5} and H = {6, 7, 8, 9}. Find the union of sets G and H. Are they disjoint?
5. 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.

6. 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 $A\cap B=\varphi$ , $A\cap$$C=\varphi$, and $B\cap C=\varphi .$

7.0Sample Questions on Disjoint Sets

1. 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 $(A\cap B=\varphi )$.

2. 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 $A\cap B=\varphi$, then the sets are disjoint. If they share any elements, they are not disjoint.

3. 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

$A\cap B=\varphi ,A\cap C=\varphi \text{, and }B\cap C=\varphi$.

4. 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.