Theory:

Important!
Click here to recall cardinality problems for two sets.
We have already the learned the cardinality of sets and practical problems for two sets. Now, let us going to solve practical problems for three sets.
For any three finite sets \(A\), \(B\) and \(C\):
 
\(n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)\)
Solving practical problems using Venn diagram
Let us say, three sets \(A\), \(B\) and \(C\) represents the employees in a company.
 
General cardinality_1.svg
 
From the above Venn diagram, we can conclude the following:
 
1. Number of employees in set \(A = a + x + z + r\)
 
2. Number of employees in set \(B = b + x + y + r\)
 
3. Number of employees in set \(C = c + y + z + r\)
 
4. Number of employees in exactly (common) three sets \(= r\)
 
5. Number of employees in set \(A\) only \(= a\)
 
6. Number of employees in set \(B\) only \(= b\)
 
7. Number of employees in set \(C\) only \(= c\)
 
8. Total number of employees in only one set \(= (a + b + c)\)
 
9. Total number of employees in only two sets \(= (x + y + z)\)
 
10. Total number of employees in at least two sets (two or more sets) \(= (x + y + z + r)\)
 
11. Total number of employees in all the three sets \(= (a + b + c + x + y + z + r)\)