22.1.4. Proof of the General Formula

Try these first

1. 1.

   What is the cardinality of the union of sets A={1,2,3} and B={2,3,4}?

   Hint

   Count the total unique elements.

2. 2.

   Define the term 'cardinality'.

   Hint

   Think about how many different items you have.

3. 3.

   What is the principle of inclusion-exclusion primarily used for?

   • Establishing set relationships
   • Counting elements in overlapping sets
   • Finding set intersections

   Hint

   Think about what complication overlaps create when counting.

4. 4.

   True or False: The cardinality of |A ∪ B| is equal to |A| + |B| only if there is no overlap.

   • True
   • False

   Hint

   Consider how overlaps affect counts.

5. 5.

   Given sets A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}. Calculate |A ∪ B ∪ C|.

   Hint

   Follow the general formula for three sets carefully.

6. 6.

   Using the principle of inclusion-exclusion, prove for n = 4 sets that |A1 ∪ A2 ∪ A3 ∪ A4| can be expanded in terms of their pairwise and higher-order intersections.

   Hint

   Ensure every intersection's contribution is correctly signed and counted.