1Look at the two Venn diagrams of the sets A, B and C, Figure 1 and Figure 2. What is the relation between n(A ∪ B ∪ C) and n(U) in each figure, and why?
2In the Venn diagram of the sets P, Q and R with their elements shown, find n(P), n(Q), n(P ∪ Q ∪ R), n_o(P), n_o(R), n(P ∩ R), the number of elements outside P ∪ Q ∪ R, n_o(P ∩ Q) and n(P ∩ Q ∩ R).
3If U = {positive integers less than 30}, P = {multiples of 2 less than 30}, Q = {multiples of 3 less than 30} and R = {multiples of 5 less than 30}, show P, Q and R in a Venn diagram and verify: n(P ∪ Q) = n(P) + n(Q) − n(P ∩ Q); the three-set union formula; and n(P ∪ Q ∪ R) = n(P − Q) + n(Q − R) + n(R − P) + n(P ∩ Q ∩ R).
4If n(U) = 100, n(M) = 45, n(E) = 50, n(S) = 35, n(M ∩ E) = 20, n(E ∩ S) = 20, n(S ∩ M) = 15 and n(M ∩ E ∩ S) = 5, find the number of elements outside M ∪ E ∪ S.
5If n(U) = 105, n(A) = 40, n(B) = 35, n(C) = 30, n(A ∩ B) = 15, n(B ∩ C) = 12, n(A ∩ B ∩ C) = 6 and 30 elements lie outside A ∪ B ∪ C, find n(A ∩ C).
6If n(U) = 120, n(M) = 50, n(E) = 40, n(S) = 45, n(M ∩ E) = 15, n(E ∩ S) = 15, n(S ∩ M) = 15 and 15 elements lie outside M ∪ E ∪ S, find n(M ∩ E ∩ S).
7If n(A ∪ B ∪ C) = 105, n_o(A) = 25, n_o(B) = 25, n_o(C) = 15, n_o(A ∩ B) = 15, n_o(A ∩ C) = 10 and n(A ∩ B ∩ C) = 10, find n_o(B ∩ C).
8Out of 90 students who sat an examination, 43 passed Science, 40 Mathematics and 38 Nepali. Of them, 13 passed Science and Mathematics, 18 Mathematics and Nepali, and 16 Science and Nepali. Show the information in a Venn diagram and find the number of students who did not pass any subject.
9In a survey of a group, 60 like tea, 45 like coffee, 30 like milk, 25 like coffee and tea, 20 like milk and tea, 15 like coffee and milk and 10 like all three drinks. Show the information in a Venn diagram and find how many people were surveyed.
10In a survey of 60 students, 23 played volleyball, 15 basketball and 20 cricket. 7 played volleyball and basketball, 5 basketball and cricket, 4 volleyball and cricket, and 15 played none of the games. Show this in a Venn diagram, find how many played all three games, and how many played only volleyball and cricket.
11In an examination, 40% passed Science, 45% Mathematics and 50% Nepali. 10% passed Science and Mathematics, 20% Mathematics and Nepali, 15% Science and Nepali, and 5% failed all three. Find the percentage who passed all three, only one, only two and at least one subject, and show this in a Venn diagram.
12In a community survey about the newspapers Yubamanch, Madhupark and Muna: 30 read Yubamanch, 25 read Madhupark, 15 read both Yubamanch and Muna, 12 read both Yubamanch and Madhupark, 9 read Madhupark only, 11 read Muna only, 5 read Yubamanch and Madhupark only, and 10 read none. Show this in a Venn diagram and find the total surveyed, the number who read exactly two newspapers and the number who read Muna.
13In a survey of 90 people about films, 48 like Nepali, 40 English and 31 Hindi. 24 like Nepali and English, 19 Hindi and English, 6 all three, and 21 like none. How many like both Nepali and Hindi films? How many do not like Hindi films? How many like neither Nepali nor Hindi films?
14In a survey of 45 people, 25 speak Nepal Bhasa, 23 speak Tamang and 15 speak Maithili. 12 speak Nepal Bhasa and Tamang, 5 Nepal Bhasa and Maithili, 10 Tamang and Maithili, and 4 speak all three. Show this in a Venn diagram. How many speak none of these languages? How many speak only one? How many speak Nepal Bhasa and Tamang but not Maithili?
15Why is \( n(A \cap B \cap C) \) added, and not subtracted, at the end of the union formula?
16In a survey of 200 people, 90 watch football, 80 watch cricket and 60 watch volleyball. 30 watch football and cricket, 25 cricket and volleyball, 20 football and volleyball, and 10 watch all three. How many watch none of the games, and how many watch exactly one?
17What is the difference between \( n(A \cap B) \) and \( n_o(A \cap B) \)? Give an example.