Rules of Operation on Sets

1. Both the Operations of Union and Intersection are Commutative. That is, given any 2 Sets \(A\) and \(B\)
   
   \(A \cup B = B \cup A\)
   
   \(A \cap B = B \cap A\)
2. Both the Operations of Union and Intersection are Associative. That is, given any 3 Sets \(A\), \(B\) and \(C\)
   
   \((A \cup B ) \cup C = A \cup (B \cup C)\)
   
   \((A \cap B ) \cap C = A \cap (B \cap C)\)
3. Union Operation Distributes over Intersection Operation. For example, given any 3 Sets \(A\), \(B\) and \(C\)
   
   \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)
4. Intersection Operation Distributes over Union Operation. For example, given any 3 Sets \(A\), \(B\) and \(C\)
   
   \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)
5. Union of Union of Sets with any of the Participating Set is the Union of Sets. For example, given any 3 Sets \(A\), \(B\) and \(C\)
   
   \(A \cup (A \cup B \cup C) = (A \cup B \cup C)\)
   
   \(B \cup (A \cup B \cup C) = (A \cup B \cup C)\)
   
   \(C \cup (A \cup B \cup C) = (A \cup B \cup C)\)
6. Intersection of Intersection of Sets with any of the Participating Set is the Intersection of Sets. For example, given any 3 Sets \(A\), \(B\) and \(C\)
   
   \(A \cap (A \cap B \cap C) = (A \cap B \cap C)\)
   
   \(B \cap (A \cap B \cap C) = (A \cap B \cap C)\)
   
   \(C \cap (A \cap B \cap C) = (A \cap B \cap C)\)
7. Intersection of Union of Sets with any of the Participating Set is the Participating Set. For example, given any 3 Sets \(A\), \(B\) and \(C\)
   
   \(A \cap (A \cup B \cup C) = A\)
   
   \(B \cap (A \cup B \cup C) = B\)
   
   \(C \cap (A \cup B \cup C) = C\)
8. Union of Intersection of Sets with any of the Participating Set is the Participating Set. For example, given any 3 Sets \(A\), \(B\) and \(C\)
   
   \(A \cup (A \cap B \cap C) = A\)
   
   \(B \cup (A \cap B \cap C) = B\)
   
   \(C \cup (A \cap B \cap C) = C\)
9. The following are the 2 De Morgan's Laws of Set Theory
   1. The Complement of the Union of Sets is same as the Intersection of the Complement of Each Set. That is, given \(N\) Sets \(A_1, A_2, A_3, ..., A_N\)
      
      \(\overline{\bigcup \limits_{i=1}^N A_i}=\overline{A_1 \cup A_2 \cup A_3 \cup \cdots \cup A_N}=\overline{A_1} \cap \overline{A_2} \cap \overline{A_3} \cap \cdots \cap \overline{A_N} =\bigcap \limits_{i=1}^N \overline{A_i}\)
   2. The Complement of the Intersection of Sets is same as the Union of the Complement of Each Set. That is, given \(N\) Sets \(A_1, A_2, A_3, ..., A_N\)
      
      \(\overline{\bigcap \limits_{i=1}^N A_i}=\overline{A_1 \cap A_2 \cap A_3 \cap \cdots \cup A_N}=\overline{A_1} \cup \overline{A_2} \cup \overline{A_3} \cup \cdots \cup \overline{A_N}=\bigcup \limits_{i=1}^N \overline{A_i}\)
10. The Elements and/or Cardinality of Union of \(N\) Number of Sets can be found out using the Principle of Inclusion and Exclusion as follows
    1. Include the Elements / Add the Cardinalities of all the \(N\) Sets.
    2. Exclude the Elements / Subtract the Cardinalities of the Intersection of all Combinations of 2 Sets from the Resultant Set / Resultant Cardinatily Value of the previous step.
    3. Include the Elements / Add the Cardinalities of the Intersection of all Combinations of 3 Sets to the Resultant Set / Resultant Cardinatily Value of the previous step.
    4. Exclude the Elements / Subtract the Cardinalities of the Intersection of all Combinations of 3 Sets from the Resultant Set / Resultant Cardinatily Value of the previous step.
    5. Continue till \(K \leq N\) to Include the Elements / Add the Cardinalities of the Intersection of all Combinations of \(K\) Sets to the Resultant Set / Resultant Cardinatily Value of the previous step if \(K\) is Odd and Exclude the Elements / Subtract the Cardinalities of the Intersection of all Combinations of \(K\) Sets from the Resultant Set / Resultant Cardinatily Value of the previous step if \(K\) is Even.
    Using the Principle of Inclusion and Exclusion as explained above, following gives the formulae for Calculating the Cardinality of Union of 2 Sets \(A\) and \(B\), Union of 3 Sets \(A\), \(B\) and \(C\) and Union of 4 Sets \(A\), \(B\), \(C\) and \(D\)
    
    \(|A \cup B| = |A| + |B| - |A \cap B|\)
    
    \(|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|\)
    
    \(|A \cup B \cup C \cup D| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |A \cap D| - |B \cap C| - |B \cap D| - |C \cap D| + |A \cap B \cap C| + |A \cap B \cap D| + |A \cap C \cap D| + |B \cap C \cap D| - |A \cap B \cap C \cap D|\)