CARDINAL NUMBER OF SETS

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Some important results derived from Venn diagrams are as follows:

A, B and C are three sets.

  1. \(\displaystyle n\left( \text{A}\cup \text{B} \right)=n\left( \text{A} \right)+n\left( \text{B} \right)-n\left( \text{A}\cap \text{B} \right)\)
  2. \(\displaystyle n\left( \text{A}\cup \text{B} \right)=n\left( \text{A} \right)+n\left( \text{B} \right)\Leftrightarrow \,\) A and B are disjoint sets.
  3. \(n\left( \text{A}-\text{B} \right)=n\left( \text{A} \right)-n\left( \text{A}\cap \text{B} \right)\)
  4. \(\displaystyle n\left( \text{A}\cup \text{B}\cup \text{C} \right)=n\left( \text{A} \right)+n\left( \text{B} \right)+n\left( \text{C} \right)-n\left( \text{A}\cap \text{B} \right)\)\(\displaystyle -n\left( \text{B}\cap \text{C} \right)-n\left( \text{A}\cap \text{C} \right)+n\left( \text{A}\cap \text{B}\cap \text{C} \right)\)
  5. \(n\left( \text{A }\!\!’\!\!\text{ }\cup \text{B }\!\!’\!\!\text{ } \right)=n\left( U \right)-n\left( \text{A}\cap \text{B} \right)\)
  6. \(n\left( \text{A }\!\!’\!\!\text{ }\cap \text{B }\!\!’\!\!\text{ } \right)=n\left( U \right)-n\left( \text{A}\cup \text{B} \right)\)
  7. \(n\left( \text{A}\,\Delta \,\text{B} \right)=\,n\left( \text{A}\cup \text{B} \right)-n\left( \text{A}\cap \text{B} \right)\)
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