Find |X ∩ Y| using union and individual sizes: If |X ∪ Y| = 18, |X| = 8, and |Y| = 15, how many elements are in X ∩ Y?

Introduction / Context:The inclusion-exclusion principle for two sets relates the sizes of each set, their union, and their intersection.

Given Data / Assumptions:

• |X ∪ Y| = 18
• |X| = 8
• |Y| = 15

Concept / Approach:|X ∪ Y| = |X| + |Y| − |X ∩ Y|. Solve for |X ∩ Y|.

Step-by-Step Solution:18 = 8 + 15 − |X ∩ Y||X ∩ Y| = 8 + 15 − 18 = 5

Verification / Alternative check:Substitute back to verify union size remains 18.

Why Other Options Are Wrong:7, 9, 11 contradict inclusion-exclusion with given numbers.

Common Pitfalls:Forgetting to subtract the intersection when adding set sizes.

Final Answer:5