22.1.2. Extension to Three Sets

Try these first

1. 1.

   What is the cardinality of the union of sets A = {1, 2} and B = {2, 3}?

   Hint

   Remember to subtract the intersection.

2. 2.

   If A = {1, 2, 3} and B = {2, 3, 4}, what is |A ∩ B|?

   Hint

   List the common elements.

3. 3.

   What does the inclusion-exclusion principle help calculate?

   • The size of a single set
   • The size of the union of multiple sets
   • The size of intersection only

   Hint

   Think about what union means.

4. 4.

   True or False: The intersection must always be subtracted in the inclusion-exclusion principle.

   • True
   • False

   Hint

   Reflect on the role of overlaps.

5. 5.

   Given sets A, B, C with cardinalities |A|=5, |B|=7, |C|=3, |A ∩ B|=2, |A ∩ C|=1, |B ∩ C|=1, |A ∩ B ∩ C|=0, find |A ∪ B ∪ C|.

   Hint

   Follow the inclusion-exclusion formula step by step.

6. 6.

   Consider four sets P, Q, R, S where |P|=6, |Q|=5, |R|=4, |S|=3 and every intersection where two sets meet is 1, and three sets meet is 0. Calculate |P ∪ Q ∪ R ∪ S|.

   Hint

   Tally each term carefully as directed.