Intersection of Sets

A set is a collection of distinct objects or elements, grouped together, often by a common property. We use curly braces to represent a set. Each item in a set is known as a member or element. Elements are separated by using commas. If we have a list of even numbers below 10, we can make a set as: even numbers = {2, 4, 6, 8}. 
 

The intersection of sets is the common elements in the given sets. In the intersection of sets, we list only the elements that are shared by all the sets.
Example: We have two sets, set A and set B. Set A consists of students who play football, A = {Anil, John, Ram}, and set B consists of students who play cricket, B = {John, Nikhil, Santhosh, Rahul}. Now, if we need the list of students who play both football and cricket, we can form a set that contains the list. The students who play both can be written as A ∩ B = {John}, since John is the  only element in both A and B.

Intersection of Sets Symbol

The symbol for the intersection of sets is ∩. The intersection of n sets can be written as set 1 ∩ set 2 ∩ set 3 ∩ … ∩ set n. If there is no common element in the given sets, then the intersection of the sets is an empty set. 
 

The properties of intersection of sets are:

• Commutative Law

• Associative Law

• Distributive Law

• Law of Empty Set

• Law of Universal Set

• Idempotent Law

Commutative Law

Commutative law states that the order of the intersection doesn’t matter. Whatever the order is, the answer remains the same.
Rule: A ∩ B = B ∩ A
Example: Let A = {cat, dog} and B = {cat, rabbit}. 
A ∩ B = {cat} — (1)
B ∩ A = {cat} — (2)
From (1) and (2), we get that A ∩ B = B ∩ A

Associative Law

While finding the intersection of three sets, we can group any two sets first. If the resultant is grouped with the remaining set, the result will be the same. 
Rule: (A ∩ B) ∩ C = A ∩ (B ∩ C)
Example: Let A = {1, 2}, B = {2, 3}, C = {2, 4}
A ∩ B = {2}
(A ∩ B) ∩ C = {2} — (1)
B ∩ C = {2}
A ∩ (B ∩ C) = {2} — (2)
From (1) and (2), (A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributive Law

If we intersect one set with the union of two other sets, it’s the same as finding the intersection with each set separately and then taking their union. 
Rule: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Example: A = {red, blue}, B ={blue, green}, C = {blue, yellow}
B ∪ C = {blue, green, yellow}
A ∩ (B ∪ C) = {blue} — (1)
A ∩ B = {blue}
A ∩ C = {blue}
(A ∩ B) ∪ (A ∩ C) = {blue} — (2)
From (1) and (2), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Therefore, the distributive law is true.

Law of Empty Set

If we take the intersection of any set with an empty set, the result will always be an empty set. 
Rule: A ∩ ∅ = ∅
Example: A = {1, 2, 3} and ∅ = {}
A ∩ ∅ = ∅, because the empty set has no elements to share with set A.

Law of Universal Set

If we intersect any set with a universal set, the result is the original set, because the universal set has all the elements in it. 
Rule: A ∩ U = A
Example: A = {sun, moon}, U = {sun, moon, stars, sky}
A ∩ U = {sun, moon}
The intersection of set A with the universal set U gives back set A. 

Idempotent Set

Intersecting a set with itself results in the same set.
Rule: A ∩ A = A
Example: A = {10, 20}
A ∩ A = {10, 20}
Intersecting a set with itself results in the same set, therefore the idempotent law is true.
 

Follow the steps given below to find the intersection of sets:

Step 1: Look at the elements in each set.

Step 2: Find the items that are common to all the sets.

Step 3: Write the common items in a new set.

Step 4: Do the same if there are more sets to check.

Step 5: The final set is the intersection of the given sets, which shows the common elements in all sets.
 

The formula for finding the intersection of two sets is:
n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
Here, n(A) is the total number of elements in set A,
n(B) is the number of elements in set B,
n(A ∪ B) is the number of elements in A ∪ B,
n(A ∩ B) is the number of elements in A ∩ B.
 

The visual way of representing the sets and their interactions is known as a Venn diagram. Let’s see the intersection of sets using two cases:

• Intersection of two sets

• Intersection of three sets

Intersection of two sets

If A and B are two sets, the resultant set A ∩ B consists of some common elements from both sets.
If A = {1, 6, 8}, B = {5, 8}
A ∩ B = {8}
The following is the Venn diagram for the intersection of two sets:

Intersection of three sets

Intersection can be performed for n sets. If A, B, and C are three sets, then the intersection of sets is the common elements in all three sets.
If P = {6, 7, 10},
Q = {7, 9, 12}
R = {7, 15}
P ∩ Q ∩ R = {7}
The Venn diagram for three sets is given below:
 

In the real world, we often need to find the common things between groups, whether it's people, items, or any data. We use the intersection of sets in those areas. Here are some of the real-world applications where the intersection of sets is used.

• Healthcare: In healthcare, the intersection of sets helps to find out the patients with multiple health conditions. 
  Set A = {patients with diabetes}
  Set B = {patients with heart problems}
  A ∩ B consists of the list of patients who have both diabetes and heart disease.
  This information helps doctors provide specialized treatment for patients with multiple health issues.
  
  
   
• Social Media: In social media, the intersection of sets helps identify mutual friends. For example, if set A represents your friends and set B represents another person's friends, then A ∩ B gives the list of friends you both have in common.
  
  
   
• Business and Marketing: In marketing, the intersection is used to identify customers who are interested in multiple products. For example, set A is a list of customers who purchased phones, and set B is the list of customers who bought earphones. A ∩ B is used to find the customers who bought both earphones and a phone together. This information helps businesses target these customers with a bundle of offers or related promotions, increasing the chances of repeat purchases and boosting sales.