Question

Show that if A, B, and C are sets, then |A ∪ B ∪ C|=|A|+|B|+|C|−|A ∩ B| − |A ∩ C|−|B ∩ C|+|A ∩ B ∩ C|.

Answer 1

To show that if A, B, and C are sets, then |A ∪ B ∪ C|=|A|+|B|+|C|−|A ∩ B| − |A ∩ C|−|B ∩ C|+|A ∩ B ∩ C|.

Step-by-step explanation:

To show that if A, B, and C are sets, then |A ∪ B ∪ C|=|A|+|B|+|C|−|A ∩ B| − |A ∩ C|−|B ∩ C|+|A ∩ B ∩ C|, we can use the inclusion-exclusion principle.

1. Start by counting the elements in the union of the three sets: |A ∪ B ∪ C|.
2. Deduct the elements that are in the intersection of two sets: - |A ∩ B| − |A ∩ C|−|B ∩ C|.
3. Add back the elements that are in the intersection of all three sets: + |A ∩ B ∩ C|.
4. By doing this, we ensure that we don't double-count the elements that are in multiple sets.

By following these steps, we can prove that |A ∪ B ∪ C|=|A|+|B|+|C|−|A ∩ B| − |A ∩ C|−|B ∩ C|+|A ∩ B ∩ C|.