1. Problem 1: Given the numbers of students in various subject combinations, find the following using a Venn diagram for Nursing (N), Business (B), and Computer (C). 2. i. All three subjects: Given as 17 (Nursing and Business and Computer). 3. ii. Business or Nursing but not Computer: This means students in B or N excluding those in C. 4. iii. Business and Nursing but not Computer: Given as 50 (Nursing and Business) minus those in all three (17), so $50 - 17 = 33$. 5. iv. At least two subjects: Sum of students in exactly two subjects plus those in all three. 6. Exactly two subjects are: - Nursing and Business but not Computer: 33 - Business and Computer but not Nursing: $18 - 17 = 1$ - Computer and Nursing but not Business: $19 - 17 = 2$ 7. So, at least two subjects = $33 + 1 + 2 + 17 = 53$. 8. v. At most two subjects: Total students minus those in all three subjects. 9. Total students = Nursing + Business + Computer - (sum of exactly two subjects) - 2 * (all three subjects) + (all three subjects) 10. Using inclusion-exclusion: $$N + B + C - (N \cap B + B \cap C + C \cap N) + (N \cap B \cap C)$$ $$= (Nursing) + (Business) + (Computer) - (50 + 18 + 19) + 17$$ 11. But we need total students first. Since the problem does not give total, we assume total is the union of all three sets. 12. vi. Exactly two subjects: Sum of students in exactly two subjects = $33 + 1 + 2 = 36$. 13. Problem 2: Given a Venn diagram with sets P, Q, and R, and values labeled including $x$, find: 14. i. Find $x$ such that $n(P) = 87$. 15. From the diagram, $n(P) = x + (x + 3) + (2x + 4) = 87$. 16. Simplify: $$x + x + 3 + 2x + 4 = 87$$ $$4x + 7 = 87$$ $$4x = 80$$ $$x = 20$$ 17. ii. $n$ is not fully specified; assuming it means total number of elements in the universal set, sum all parts: 18. $n = n(P \cup Q \cup R) = n(P) + n(Q) + n(R) - n(P \cap Q) - n(Q \cap R) - n(P \cap R) + n(P \cap Q \cap R)$ 19. Using values: - $n(P) = 87$ - $n(Q) = 18 + x = 18 + 20 = 38$ - $n(R)$ parts: $x + 3 = 23$, plus other parts unknown, so insufficient data to find total. 20. iii. $(R)'$ is the complement of $R$, meaning elements not in $R$. Without total universal set size, cannot find exact number. 21. iv. $n(R \cup Q)$ is the number of elements in $R$ or $Q$ or both. 22. v. $n(P \cap Q)'$ is the complement of the intersection of $P$ and $Q$. 23. vi. Probability of $n(R)$ is $\frac{n(R)}{n(U)}$, where $n(U)$ is total elements in universal set. 24. Without total universal set size, exact numerical answers for parts ii, iii, iv, v, and vi cannot be determined. Final answers: - Problem 1: i. All three subjects = 17 ii. Business or Nursing but not Computer = (Nursing + Business) - (all three + Business and Computer + Nursing and Computer) = $50 + ? - (17 + 18 + 19)$ but incomplete data for exact number. iii. Business and Nursing but not Computer = 33 iv. At least two subjects = 53 v. At most two subjects = Total - 17 (all three subjects) vi. Exactly two subjects = 36 - Problem 2: i. $x = 20$ "slug":"venn diagram problems","subject":"set theory","desmos":{"latex":"","features":{"intercepts":false,"extrema":false}},"q_count":2