Theory:

Let \(A = \{20\), \(25\), \(26\), \(30\), \(31\), \(33\), \(35\}\), \(B = \{25\), \(30\), \(32\), \(34\), \(35\), \(36\}\) and \(C = \{15\), \(26\), \(30\), \(32\), \(33\), \(35\}\).
 
(i) Calculate the total number of common elements in all the three sets.
 
(ii) Calculate the total number of common elements in \(A\) and \(B\).
 
(iii) Calculate the total number of common elements in \(B\) and \(C\).
 
(iv) Calculate the total number of common elements in \(A\) and \(C\).
 
(v) Calculate the total number of elements in all the three sets.
 
Solution:
 
Cardinality_1.svg
 
\(A = \{20\), \(25\), \(26\), \(30\), \(31\), \(33\), \(35\}\)
 
Number of elements in set \(A = n(A) = 7\)
 
\(B = \{25\), \(30\), \(32\), \(34\), \(35\), \(36\}\)
 
Number of elements in set \(B = n(B) = 6\)
 
\(C = \{15\), \(26\), \(30\), \(32\), \(33\), \(35\}\)
 
Number of elements in set \(C = n(C) = 6\)
 
(i) Total number of common elements in all the three sets:
 
\(A \cap B \cap C\) \(=\) \(\{30\), \(35\}\)
 
\(n(A \cap B \cap C) = 2\)
 
(ii) Total number of common elements in \(A\) and \(B\):
 
\(A \cap B\) \(=\) \(\{25\), \(30\), \(35\}\)
 
\(n(A \cap B) = 3\)
 
(iii) Total number of common elements in \(B\) and \(C\):
 
\(B \cap C\) \(=\) \(\{30\), \(32\), \(35\}\)
 
\(n(B \cap C) = 3\)
 
(iv) Total number of common elements in \(A\) and \(C\):
 
\(A \cap C\) \(=\) \(\{26\), \(30\), \(33\), \(35\}\)
 
\(n(A \cap C) = 4\)
 
(v) Total number of elements in all the three sets:
 
\(n(A \cup B \cup C)\) \(= n(A) + n(B) + n(C)\) \(- n(A \cap B)\) \(- n(B \cap C)\) \(- n(A \cap C)\) \(+ n(A \cap B \cap C)\)
 
\(n(A \cup B \cup C)\) \(=\) \(7 + 6 + 6 - 3 - 3 - 4 + 2\)
 
\(n(A \cup B \cup C)\) \(= 11\)
 
Important!
If \(A\) and \(B\) are two finite sets, then:
 
1. \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)
 
2. \(n(A - B) = n(A) - n(A \cap B)\)
 
3. \(n(B - A) = n(B) - n(A \cap B)\)
 
4. \(n(A^{\prime}) = n(U) - n(A)\)
 
5. \(n(U) = n(A) + n(A^{\prime})\)