Inclusion-Exclusion Principle

The inclusion-exclusion principle is used to find the cardinality of the union of multiple sets by including the sizes of the sets and then excluding the sizes of their intersections.

For Two Sets

Given two sets $A$ and $B$:

1. Union of A and B: $|A \cup B| = |A| + |B| - |A \cap B|$

2. Intersection of A and B: $|A \cap B| = |A| + |B| - |A \cup B|$

3. Elements only in A (not in B): $|A \setminus B| = |A| - |A \cap B|$

4. Elements only in B (not in A): $|B \setminus A| = |B| - |A \cap B|$

5. Elements in A but not in B, plus elements in B but not in A (symmetric difference): $|A \Delta B| = |A \setminus B| + |B \setminus A| = |A| + |B| - 2|A \cap B|$

For Three Sets

Given three sets $A$, $B$, and $C$:

1. Union of A, B, and C: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$

2. Intersection of A and B only (excluding C): $|A \cap B \setminus C| = |A \cap B| - |A \cap B \cap C|$

3. Intersection of A and C only (excluding B): $|A \cap C \setminus B| = |A \cap C| - |A \cap B \cap C|$

4. Intersection of B and C only (excluding A): $|B \cap C \setminus A| = |B \cap C| - |A \cap B \cap C|$

5. Elements only in A (not in B or C): $|A \setminus (B \cup C)| = |A| - |A \cap B| - |A \cap C| + |A \cap B \cap C|$

6. Elements only in B (not in A or C): $|B \setminus (A \cup C)| = |B| - |A \cap B| - |B \cap C| + |A \cap B \cap C|$

7. Elements only in C (not in A or B): $|C \setminus (A \cup B)| = |C| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$

8. Elements in A and B, but not in C: $|(A \cap B) \setminus C| = |A \cap B| - |A \cap B \cap C|$

9. Elements in A and C, but not in B: $|(A \cap C) \setminus B| = |A \cap C| - |A \cap B \cap C|$

10. Elements in B and C, but not in A: $|(B \cap C) \setminus A| = |B \cap C| - |A \cap B \cap C|$

11. Elements in exactly one of the sets: $|A \setminus (B \cup C)| + |B \setminus (A \cup C)| + |C \setminus (A \cup B)|$

12. Elements in exactly two of the sets: $|(A \cap B \setminus C)| + |(A \cap C \setminus B)| + |(B \cap C \setminus A)|$

Would you like further details or explanations on any of these formulas? Here are some additional questions you might consider asking:

1. How do you apply the inclusion-exclusion principle in probability?
2. Can you provide an example problem using the inclusion-exclusion principle for two sets?
3. How do you extend the inclusion-exclusion principle to more than three sets?
4. What are some common applications of the inclusion-exclusion principle?
5. How do you visualize the inclusion-exclusion principle using Venn diagrams?

Tip: When working with the inclusion-exclusion principle, drawing Venn diagrams can help visualize the overlaps and intersections between sets, making it easier to apply the formulas correctly.