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# 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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