# Question Why does the Inclusion-Exclusion Principle require subtracting the intersection when computing $|A \cup B|$? ## Answer Choices (A) To ensure the total count equals the universal set (B) To avoid counting elements that appear in both sets twice (C) To handle cases when sets are disjoint (D) To make the calculation symmetric (E) To account for elements only in one set # Solution The Inclusion-Exclusion Principle for two sets states: $|A \cup B| = |A| + |B| - |A \cap B|$ The reason we subtract $|A \cap B|$ is to correct for double-counting. When we add $|A|$ and $|B|$, we count every element in A and every element in B. However, elements in the intersection $A \cap B$ belong to both sets, so they get counted twice in the sum $|A| + |B|$. To get the correct count of distinct elements in $A \cup B$, we must subtract $|A \cap B|$ once to compensate for this double-counting. Without this subtraction, we would overcount by exactly the number of elements in the intersection. **Answer: (B)**