A ∩ B ∩ C

The intersection of sets refers to the collection of elements common to all the given sets. For example, set A students who take German and set B students who take Japanese, the intersection of the sets of students who take German and those who take Japanese includes only students who are enrolled in both language classes.

Properties of Intersection of Sets

The following are some properties of the intersection operation:

Commutative law: A ∩ B=B ∩ A

Examine the two sets A = {5, 6, 7, 8, 9, 10} and B = {6, 7, 9, 11}
Here, A ∩ B = {5, 6, 7, 8, 9, 10} ∩ {6, 7, 9, 11} = {6, 7, 9}
B ∩ A = {6, 7, 9, 11} ∩ {5, 6, 7, 8, 9, 10} = {6, 7, 9}
Hence, A ∩ B=B ∩ A

Associative law: (A ∩ B) ∩ C=A ∩ (B ∩ C)
Examine the three sets A = {5, 6, 7, 8}, B = {7, 8, 9, 10}, and C = {9, 10, 11, 12}.
Now,
A ∩ B={5, 6, 7, 8} ∩ {7, 8, 9, 10}={7,8}
(A ∩ B) ∩ C = {7,8} ∩ {9, 10, 11, 12}={}=φ (No elements are intersecting, so it is an empty set.)
Now, let us find A ∩ (B ∩ C)
B ∩ C={7, 8, 9, 10} ∩ {9, 10, 11, 12}={9, 10}
A ∩ (B ∩ C)= {5, 6, 7, 8} ∩ {9, 10}={}=φ (No elements are intersecting, so it is an empty set.)
Hence, (A ∩ B) ∩ C=A ∩ (B ∩ C) 

Law of φ and U: φ ∩ A=φ, U ∩ A=A 
Now examine, φ = {} and A = {12, 14, 11}
φ ∩ A={} ∩ {12, 14, 11}={}=φ (The intersection with the empty set gives the empty set. 
Let U = {3, 5, 7, 9, 11, 15, 17, 19, 21, 25} and A = {5, 9, 15, 19, 25}. Then,
U ∩ A={3, 5, 7, 9, 11, 15, 17, 19, 21, 25} ∩ {5, 9, 15, 19, 25}= {5, 9, 15, 19, 25}=A
The intersection with the universal set gives back the original set A, since all its elements are in U.

Idempotent law: A ∩ A=A
If A={e, f, g, h, i} 
Therefore, A ∩ A={e, f, g, h, i} ∩ {e, f, g, h, i}={e, f, g, h, i}=A

Distributive law: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Consider the following three sets: A =  {1, 3, 5, 7}, B =  {1, 2, 4, 6}, and 
C =  {2, 7, 4, 8}.
B ∪ C={1, 2, 4, 6} ∪ {2, 7, 4, 8}={1, 2, 4, 6, 7, 8}
Next, A ∩ (B ∪ C)={1, 3, 5, 7} ∩ {1, 2, 4, 6, 7, 8}={1, 7} 
 A ∩ B={1} and A ∩ C={7} 
(A ∩ B) ∪ (A ∩ C)={1} ∪ {7}={1,7} 

Intersection of Sets Venn Diagram

The intersection of two sets, A and B, is shown by the shaded area in the diagram above. Likewise, as an example below, we can create a Venn diagram for the intersection of three sets.
 

The expression A ∩ B ∩ C represents the set of elements that are common to all three sets, A, B, and C.
For example, let us assume that the three sets are A={b, c, d, e}, B={c, d, e, f, g}, and C={d, e, f, g, h, i}. Now, let us examine the intersection of two sets, one at a time.
A ∩ B={c, d, e}
B ∩ C={d, e, f, g}
A ∩ B ∩ C=(A ∩ B) ∩ (A ∩ C)
= {c, d, e} ∩ {d, e, f, g}
= {d,e}
Here, we used the basic rule that says the intersection of two sets includes the elements they have in common. Using this, the elements that are shared between A, B, and C are. 

How to find A ∩ B ∩ C?

The following steps make it simple to calculate the A ∩ B ∩ C.

• The first step is to determine where the elements of sets A and B intersect. This is shown as A and B.
• Secondly, let us look for the elements that are present in both B and C. The symbol for this is B ∩ C.
• Finding the point where the two results of the intersection of elements intersect is the third and last step. Here, we have to check A and B as well as B and C. And finally, we have the solution for intersections A, B, and C. This is simply expressed as follows: A ∩ B ∩ C=(A ∩ B) ∩ (B ∩ C) 
   

 A ∩ B ∩ C shows the overlap between the three sets. Let us see how A ∩ B ∩ C, helps in real-life situations that involve multiple criteria.

• Verification of student records
  Schools compare groups of students who attended the orientation (C), paid the fee (A), and enrolled in the course (B). Students who fulfilled all necessary requirements are represented by A ∩ B ∩ C, guaranteeing complete compliance and valid enrollment. 
• Health program eligibility 
  Patients who have insurance (A), have finished their physical (B), and have signed up for a wellness program (C) are all assessed by hospitals. Patients who are eligible for all three advanced care packages or discounts are included in A ∩ B ∩ C. This means that they meet the conditions of sets A, B, and C.
• E-commerce targeting  
  Users who visited the product page (A), added the item to their cart (B), and completed a purchase (C) are tracked by online platforms. High-conversion clients are identified for loyalty rewards and tailored advertisements by A ∩ B ∩ C.
• Employee appraisal shortlist
  Employees who completed projects on time (A), hit performance goals (B), and participated in all training sessions (C) are shortlisted by employers. The list of top-performing workers who qualify for bonuses or promotions is provided by A ∩ B ∩ C.
• College admission process 
  Applicants who complete the admission process, who have completed the application form (is said to be A), uploaded the required documents (is said to be B), and paid the application fee (is said to be C) are considered by the universities. Applicants in A, B, and C have finished the process, and they are prepared for evaluation.