1Fifteen students with roll numbers 1 to 15 were asked whether they like Mathematics or Science. Roll numbers 1, 3, 4, 11 and 13 like Mathematics only; 2, 8 and 15 like both; 5, 6, 7, 10 and 12 like Science only; 9 and 14 like neither. Using the listing method, write the sets of students who like Mathematics, Mathematics only, Science, Science only, both and neither, each with its cardinal number. How many students were surveyed?
2In grade 10, 15 students like coffee, 10 like tea and 6 like both; 5 dislike both. How is this shown in a Venn diagram? How many like coffee only and how many like tea only? How many students are in the class?
3Present the cardinality of sets with examples.
4For two sets A and B with \( A \subset B \), find \( n(A \cup B) \) and \( n(A \cap B) \).
5If A and B are overlapping sets, state the formula for \( n(A \cup B) \).
6Sets A and B have 12 and 8 elements respectively. Find the minimum number of elements that could be in \( A \cup B \).
7A Venn diagram of sets M and E has 20 in M only, 60 in both, 30 in E only and 15 outside both circles, so 80 people are in M, 90 are in E and 15 are in neither. Find \( n_0(M) \), \( n_0(E) \), \( n(M) \), \( n(E) \), \( n(M \cup E) \), \( n(M \cap E) \), \( n(\overline{M \cup E}) \) and \( n(U) \).
8If \( n(U) = 200 \), \( n_0(M) = 2x \), \( n_0(E) = 3x \), \( n(M \cap E) = 60 \) and \( n(\overline{M \cup E}) = 40 \), find the value of x.
9If \( n(U) = 350 \), \( n(A) = 200 \), \( n(B) = 220 \) and \( n(A \cap B) = 120 \), find \( n(A \cup B) \) and \( n(\overline{A \cup B}) \).
10If \( n(A) = 35 \) and \( n(\overline{A}) = 25 \), find the value of \( n(U) \).
11Of two sets P and Q, P has 40 elements, \( P \cup Q \) has 60 elements and \( P \cap Q \) has 10 elements. How many elements are in Q?
12In a survey of 180 students of a school, 45 like Nepali only, 60 like English only and 15 like neither subject. Show this in a Venn diagram, and find how many like both subjects and how many like at least one.
13Among 1200 students of a school, 100 like Mathematics only, 200 like Science only and 700 like neither subject. Show this in a Venn diagram, and find how many like both subjects and how many like at least one.
14Among 60 students, 10 play football only, 20 play volleyball only and 12 play neither game. Show this in a Venn diagram, and find how many play both games and how many play at least one.
15Among 900 people of a community, 525 read Madhupark, 450 read Yubamanch and 75 read neither newspaper. Show this in a Venn diagram, and find how many read both and how many read only one newspaper.
16Among 150 people, 90 like modern songs, 70 like folk songs and 30 like neither. Show this in a Venn diagram, and find how many like both and how many like only modern songs.
17Among 360 players, 210 like to play volleyball, 180 like to play football and 30 like neither. Show this in a Venn diagram, and find how many like both games and how many like only one game.
18Of the students in an examination, 70% passed English, 60% passed Mathematics, 20% failed both and 550 passed both. Show this in a Venn diagram, and find the total number of students and the number who passed English only.
19Of the students who sat the grade 10 examination, 60% are interested in studying Science, 70% in Management, 10% in neither, and 400 in both. Show this in a Venn diagram, and find the number of students surveyed and the number interested in Science only.
20In a community, 65% ride a motorcycle, 35% ride a scooter and 20% ride both, and the people who ride both number 200. Show this in a Venn diagram, and find how many people took part in the survey and how many ride a motorcycle only.
21Among 95 people of a community, the numbers who drink tea and coffee are in the ratio 4 : 5, 10 drink both and 15 drink neither. Show this in a Venn diagram, and find how many drink exactly one and how many drink at least one.
22In a class of 64 students, the numbers who like milk only and curd only are in the ratio 2 : 1, and 16 like both. Show this in a Venn diagram, and find how many like milk and how many like only one kind of drink.
23At a conference of 320 participants, 60 only sing and 100 only dance. The number who do neither is three times the number who do both. Show this in a Venn diagram, and find how many do neither and how many do at most one.
24Among 200 people of a community, the numbers who use a laptop only and a mobile only are in the ratio 2 : 3, 30% use both and 15% use neither. Show this in a Venn diagram, and find how many use a laptop and how many use at most one gadget.
25Of 300 players, one third play volleyball only. 60% of the remaining players play football only, and 60 play neither game. Using a Venn diagram, find the ratio of the number who play volleyball to the number who play football.
26Among 65 players in a survey, 11 play volleyball only and 33 play cricket only. If the number who play cricket is double the number who play volleyball, use a Venn diagram to find how many play both and how many play neither.
27In a survey of 80 people, 60 like oranges and 10 like both oranges and apples. The number who like oranges is 5 times the number who like apples. Using a Venn diagram, find the number who like apples only and the number who like neither fruit.
28Why is \( n(A \cap B) \) subtracted in the formula for \( n(A \cup B) \)?
29In a class of 40, 25 like singing, 20 like dancing and 8 like neither. Find how many like exactly one and how many like at most one.
30Can \( n(A \cup B) \) ever be smaller than \( n(A) \)? Give a reason.