1. **State the problem:** We have 12 pupils in a class. They are asked if they have a brother or a sister. 2. **Given data:** - Total pupils $|\varepsilon| = 12$ - Pupils with a brother $|B| = 9$ - Pupils with a sister $|S| = 7$ - Pupils with neither brother nor sister $|\varepsilon - (B \cup S)| = 2$ 3. **Find:** The number of pupils in each section of the Venn diagram: only brother, only sister, and both. 4. **Use the formula for union of two sets:** $$|B \cup S| = |B| + |S| - |B \cap S|$$ 5. Since 2 pupils have neither, the number in $B \cup S$ is: $$|B \cup S| = |\varepsilon| - 2 = 12 - 2 = 10$$ 6. Substitute values into the union formula: $$10 = 9 + 7 - |B \cap S|$$ 7. Solve for the intersection: $$|B \cap S| = 9 + 7 - 10 = 16 - 10 = 6$$ 8. Find pupils with only a brother: $$|B| - |B \cap S| = 9 - 6 = 3$$ 9. Find pupils with only a sister: $$|S| - |B \cap S| = 7 - 6 = 1$$ 10. **Final Venn diagram values:** - Only brother: 3 - Only sister: 1 - Both brother and sister: 6 - Neither: 2 This matches the given numbers in the problem. **Answer:** Only brother = 3, Only sister = 1, Both = 6, Neither = 2