CARDINAL NUMBER OF SETS

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)

Post Author: E-Maths