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# Practice Paper 2 Question 4

Three planar regions $$A$$, $$B$$, $$C$$ partially overlap each other, with $$|A| = 90,$$ $$|B| = 90,$$ $$|C| = 60$$ and $$|A \cup B \cup C| = 100,$$ where $$|\cdot|$$ denotes the area. Find the minimum possible $$|A \cap B \cap C|$$.

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## Hints

• Hint 1
Try to find the minimum $$|A\cap B|.$$
• Hint 2
... by considering the maximum $$|A\cup B|.$$
• Hint 3
What does that minimal case imply about $$C$$ in relation to $$A$$ and $$B,$$ given the question conditions?
• Hint 4
... more specifically, in relation to $$A\cup B,$$ given $$|A \cup B \cup C| = 100?$$
• Hint 5
Given $$C$$ must be contained within $$A\cup B,$$ what does that imply about $$C$$ in relation to $$A\cap B?$$
• Hint 6
... more specifically, how must $$C$$ be distributed outside of $$A\cap B$$ in order to minimize the full intersection?

## Solution

First consider the minimum size of $$A \cap B$$. We know that $$|A \cup B|=|A|+|B|-|A\cap B|$$ and since $$|A \cup B\cup C|=100$$ then $$|A \cup B|\le100$$ so $$|A \cap B|\ge80$$. In this minimal case, $$C$$ must be contained within $$A \cup B,$$ because otherwise $$|A \cup B \cup C| > 100$$. We want to have as much of $$C$$ as possible outside of $$A\cap B,$$ hence the minimal size of the intersection is $$|C| - (|A \cup B| - |A \cap B|)=40$$.

Note: Drawing standard (circular) Venn diagrams for $$A,B,C$$ to meet the above conditions is not possible, but it is possible using other shapes. Give it a try.

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