Out of 90 students who participated in an examination, 43 passed in Science, 40 in Mathematics and 38 in Nepali. Among them, 13 passed in Science and Mathematics, 18 in Mathematics and Nepali as well as 16 passed in Science and Nepali. Using the information, answer the following questions: i) Show the information in Venn diagram. ii) Find the number of students who did not pass in any subject.

In 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. Based on the information, answer the following questions: i) Show the information in a Venn-diagram. ii) Find how many people were surveyed.

In a survey among 60 students, 23 played volleyball, 15 played basketball and 20 played cricket. If 7 played volleyball and basketball, 5 played basketball and cricket, 4 played volleyball and cricket but 15 played neither of the games. Based on this information, answer the following questions: i) Show the information in Venn-diagram. ii) Find how many students played all the three games. iii) How many played only volleyball and cricket.

In a survey of 125 students of class 10 in a school, 65 preferred to visit Lumbini, 75 preferred to visit Pokhara and 25 preferred both the places in their annual excursion.

(i) Show the above information in a Venn-diagram
(ii) Find the number of students who liked neither of two places.
(iii) Find the number of students who liked only Lumbini.

In a survey of 2000 Indian tourists who arrived in Nepal, 65% wished to visit Pashupati, 50% wished to visit Chandragiri and 45% wished to visit Manakamana. Similarly, 35% wished to visit Pashupati and Chandragiri, 25% to Chandragiri and Manakamana, and 20% to Manakamana and Pashupati. If 5% wished to visit none of these places, find the number of tourists who wished to visit all these three places.

In a survey of 300 people, it was found that 150 people like iPhones and 200 people like Android phones. But 25 people did not like any of these two phones.

1. If I and A denote the sets of people who like iPhone and Android Phones respectively, write the cardinality of $\rm n \overline {( I \cup A ) }$.
2. Present the above information in a Venn diagram.
3. Find the number of people who like iPhones only.
4. Compare the number of people who like both iPhone and Android phones and who do not like any of these two phones.

If n(A) = 35, n(B) = 30, and n(A $\rm \cap$ B) = 20, find n (A $\cup$ B) and n_o(A) by using a Venn diagram.

If n(X) = 40, n(Y) = 60 and n(X $\rm \cup$ Y) = 85, then find n(X $\rm \cap$ Y) and n_o(Y) by using formula.

If P = {Multiples of 2 up to 20}, Q = {Multiples of 3 up to 24} and U = {integers from 1 to 25}, find n_o(P) and n(P $\rm \cup$ Q) by using a Venn diagram.

If A = {factors of 18}, B = {multiples of 3 up to 27}, and U = {integers from 1 to 30}, find the cardinality of n_o(B) and n(A$\rm \cup$B) by using a Venn diagram.

In sets A and B, A has 40 members, B has 50 members, and $\rm (A \cup B)$ has 60 members. By how many elements $\rm (A \cap B)$ is formed?

In sets A and B, A has 50 members, B has 60 members, and 30 members are the same in both sets. By how many elements $\rm (A \cup B)$ is formed?

Out of 90 civil servants, 65 were working in the office, 50 were working in the field and 35 were working in both the premises (sites).

1. Let O and F represent the set of civil servants working in office and field respectively, then find $\rm n(F)$ and $\rm n(O \cap F)$.
2. Represent the given information in a Venn diagram.
3. How many civil servants were absent?
4. How many civil servants were working in the field only?

Out of 100 students, 80 passed in Science, 71 in Mathematics, 10 failed in both subjects, and 7 did not appear in an examination.

1. Let S and M represent the sets of students who were passed in science and Maths respectively, then find n(S) and n(M).
2. Find the number of students who passed in either Science or Mathematics.
3. Find the number of students who passed in both subjects.
4. Represent the above information in a Venn diagram.

In an examination, it was found that 55% failed in Maths and 45% failed in English. If 35% passed in both subjects

1. What percent failed in Maths only?
2. What percent failed in English only?
3. Represent the above information in a Venn diagram.
4. Find the percentage of students who failed in both subjects.

45% of the students of a school play basketball, 40% play cricket, and 30% play both. If 360 students play neither basketball nor cricket then answer the following questions:

1. If B and C represent the sets of students who like basketball and cricket, respectively, what percentage represents $\rm n \overline { \left ( C \cup B \right )} $ in the question?
2. Find the number of students who play either basketball or cricket.
3. Find the total number of students.
4. Find the number of students who play basketball only.

In an exam, 70% of the examinees passed in Science, 75% in Maths, 10% of them failed in both subjects, and 220 examinees passed in both subjects. If S and M are the set of examinees passed in Science and Maths respectively, then answer the following questions:

1. Find the percentage of n(S $\rm \cup$ M).
2. FInd the percentage of n(S $\rm \cap$ M).
3. Find the total number of students.
4. What percentage in the Venn diagram has been covered by n(S $\rm \triangle$ M)?

The population of the village is 15000. Among them 9000 read Magazine A, 7500 read Magazine B, and 40% read both magazines.

1. Find the percentage of $\rm n(A \cap B)$.
2. Find the value of $\rm n(A \cup B)$.
3. Find the number of people who don't read both magazines.
4. Find the percent of people who don't read both magazines.

In a class of 25 students, 12 have chosen Mathematics, 8 have chosen Mathematics but not Biology. If each of them has chosen at least one, then

1. What is the relation between the total number of students and the students who have not chosen Mathematics or Biology?
2. Find the number of students who have chosen both Mathematics and Biology.
3. Find the number of students who have chosen Mathematics but not Biology.
4. Find the cardinality of the set representing the symmetric difference of the set of the students who like Mathematics and Biology.

In a class of 65 students, 10 students liked Maths but not English, and 20 students liked English but not Maths. If 5 students did not like both then,

1. Find the number of students who liked Maths or Science.
2. Find the number of students who liked both Maths and Science.
3. Show the given information in a Venn-diagram.
4. Find the cardinality of the set representing the symmetric difference of the set of students who like Maths and Science.

Out of 100 students of class V, 73 passed in Mathematics and 84 in Nepali in the final examination but 7 failed in both subjects and 5 were absent in the examination.

1. If M and N represent the set of students who passed in Mathematics and Nepali then what are the values of n(M) and n(N)?
2. Find the total number of students who failed in both subjects.
3. Find the number of students who passed in either of the subjects.
4. Find the number of students who passed in both subjects.
5. Show the given information in a Venn-diagram. Which region in the Venn-diagram represents the maximum number of students?

Out of 75 students of class X, 30 passed in Mathematics and 40 in Social Studies in the final examination, but 10 failed in both subjects and 5 were absent in the examination.

1. If M and S represent the set of students who passed in Maths and Social Studies, find the value of n(M) and n(S).
2. Find the total number of students who failed in both subjects.
3. Find the number of students who passed in both subjects.
4. Show the given information in a Venn-diagram. Which region in the Venn-diagram represents the minimum number of students?

Out of 100 students in a class, 20 students like Maths but not Science, and 30 students like Science but not Maths. If 20 students like neither of the subjects

1. Let M and S represent the set of students who like Maths and Science respectively then write the value of $\rm n_o(M)$ and $\rm n_o(S)$.
2. Show the given information in a Venn-diagram.
3. Find the number of students who like both Maths and Science.
4. Find the ratio of the students who like Maths to the students who like Science. 

250 students in a group were asked whether they like mango or apple. 80 students like mango but not apple and 50 students like apple but not mango. If 50 students do not like both of the fruits, then,

1. Let M and A represent the set of students who like mangoes and apples respectively then write the value of $\rm n_o(M)$ and $\rm n_o(A)$.
2. Show the given information in a Venn diagram.
3. Find the number of students who like both mangoes and apples.
4. Find the ratio of the students who like mango to the students who like apple.

1000 students in a school like Nepali, History, or both subjects. Out of them, 400 like both subjects. If the ratio of the number of students who like Nepali and History is 3:2, by using the Venn diagram, find

1. Find the number of students who like Nepali.
2. Find the number of students who like History.
3. Find the number of students who like only one subject.
4. Show the results in a Venn diagram.

In a school, 640 teachers like either milk or curd or both. The ratio of the number of teachers who like milk to the number of teachers who like curd is 3:2 and 160 teachers like both milk and curd. Find:

1. Find the number of students who like milk.
2. Find the number of students who like curd.
3. Find the number of students who like only one: milk or curd.
4. Show the results in a Venn diagram.

If n(A) = 40, n(B) = 60, and n(A $\rm \cup$ B) = 80,

1. Find the value of n (A $\rm \cap$ B).
2. Draw a Venn diagram of the above information.

A and B are two subsets of a universal set U in which n(U) = 43, n(A) = 25, n(B) = 18, and n(A $\rm \cap$ B) = 7.

1. Draw a Venn diagram of the above information.
2. Find the value of $\rm n \overline{ A \cup B} $.

If A and B are two subsets of a universal set U in which, n(U) = 70, n(A) = 40, n(B) = 20, and $\rm n \overline { A \cup B}$ = 15 then

1. Show the above information in a Venn diagram.
2. Find the value of $\rm n ( A \cap B)$.

If n(A) = 40, n(B) = 60 and $\rm n (A \cup B) = 80$, then

1. find the value of $\rm n( A \cap B)$.
2. find the value of $\rm n_o (A)$.
3. show it in a Venn Diagram.

In a survey of 60 students, 30 drink milk, 25 drink curd, and 10 students drink milk as well as curd.

1. If the set of total students is U, then write the cardinality of U.
2. Draw a Venn diagram to illustrate the above information.
3. Find the number of students who drink either of them.
4. Find the number of students who drink neither of them.

A survey of a community shows that 55% of the people like to listen to the radio, 65% like to watch the television, and 35% like to listen radio as well as to watch the television.

1. If the set of people in the community is U then write the cardinality of U in percentage.
2. Show the above information in a Venn-diagram.
3. Find the percentage of people who like either to listen to the radio or to watch the television.
4. Find the percentage of people who do not like to listen to the radio as well as to watch the television.

In a class of 30 students, 20 students like to play cricket and 15 like to play volleyball. Also, each student likes to play at least one of the two games.

1. If C and V be the set of students who like cricket and volleyball respectively then find the cardinality of C and V.
2. Illustrate the above information in a Venn diagram.
3. Find the number of students who like either of the games.
4. How many students like to play both the games?

In a survey before an election, 65% of people liked leader A and 60% of people liked leader B. If 15% of people did not like to open their opinion about any of the leaders,

1. If A and B denote the set of people who like leader A and leader B respectively and n(U) = 100 then find n(A) and n(B).
2. Show the given information in a Venn-diagram.
3. Find the percentage of people who liked both the leaders by using the Venn diagram.
4. Find the percentage of people who like either of the leaders.

In a group of 100 students, 68 liked the football game, 60 liked the volleyball game and 2 didn't like any of the games.

1. Shwo the given information in a Venn diagram.
2. How many students like both games?
3. How many students like only football?
4. How many students like either of the games?

In a survey of 119 students, it was found that 16 drink neither milk nor tea and 69 drink milk and tea.

1. Show the given information in a Venn diagram.
2. How many students drink milk only?
3. How many students drink tea only?
4. How many students like either of the drinks?

If A = {a,c,e}, B = {b,c,d}, and C = {a,c,d,f}, find $\rm n(A \cap B \cap C)$.

If A = {1, 2, 3}, B = {2, 3, 4}, and C = {3, 4, 5}, find $\rm n(A \cup B \cup C)$.

If n(A) = 65, n(B) = 50, n(C) = 35, n(A $\rm \cap$ B) = 25, n(B $\rm \cap$ C) = 20, n(C $\rm \cap$ A) = 15, n(A $\rm \cap$ B $\rm \cap$ C) = 5, and n(U) = 100,

1. Are the sets A, B, and C overlapping sets? Give reason.
2. Find the value of $\rm n(A \cup B \cup C)$.
3. Find the value of $\rm n \overline{ A \cup B \cup C }$.
4. Show the given information in a Venn diagram.

If n(A) = 48, n(B) = 51, n(C) = 40, n(A $\rm \cap$ B) = 11, n(B $\rm \cap$ C) = 10, n(C $\rm \cap$ A) = 9, n(A $\rm \cap$ B $\rm \cap$ C) = 4 and n(U) = 120,

1. Are the sets A, B, and C overlapping sets? Give reason.
2. Find the value of n($\rm A \cup B\cup C$).
3. Find the value of n($\rm \overline{A \cup B \cup C}$).
4. Present the above information in a Venn diagram.

Sets A, B, and C are the subsets of the universal set U. If n(U) = 300, n(A) = 100, n(B) = 90, n(C) = 110, n(A$\rm \cap$B) = 60, n(B$\rm \cap$C) = 40, n(C$\rm \cap$A) = 45, and n(A $\rm \cup$ B $\rm \cup$ C) = 200.

1. The sets A, B, and C are overlapping sets. Give reason.
2. Find the value of n(A $\rm \cap$ B $\rm \cap$ C).
3. Find the value of n($\rm \overline{A \cup B \cup C}$).
4. Present the given information in a Venn diagram.

A, B and C are the subsets of the universal set U. If n(U) = 100, n(A) = 60, n(B) = 45, n(C) = 30, n(A $\rm \cap$ B) = 25, n(B $\rm \cap$ C) = 20, n(C $\rm \cap$ A) = 15, n(A $\rm \cup$ B $\rm \cup$ C) = 85.

1. The sets A, B, and C are overlapping sets. Give reason.
2. Find the value of n(A $\rm \cap$ B $\rm \cap$ C).
3. Find the value of n($\rm \overline{A \cup B \cup C}$).
4. Present the given information in a Venn diagram.

If n(A) = 12, n(B) = 12, n(A $\rm \cap$ B) = 5, n(A $\rm \cap$ C) = 3, n(B $\rm \cap$ C) = 4, n(A $\rm \cap$ B $\rm \cap$ C) = 2 and n(A $\rm \cup$ B $\rm \cup$ C) = 20, then

1. Find the value of $\rm n_{o}(C)$.
2. Find the value of $\rm n(C)$.
3. Find the value of $\rm n(A \cup C)$.
4. Present in a Venn diagram.

If n(A) = 36, n(B) = 36, n(A $\rm \cap$ B) = 15, n(A $\rm \cap$ C) = 15, n(B $\rm \cap$ C) = 12, n(A $\rm \cup$ B $\rm \cup$ C) = 66, then

1. Find the value of $\rm n_{o}(C)$.
2. Find the value of $\rm n(C)$.
3. Find the value of $\rm n(A \cup C)$.
4. Present in a Venn diagram.

If n(A) = 14, n(B) = 13, n(C) = 22, n(A $\rm \cap$ B $\rm \cap$ C) = 4, n(A $\rm \cap$ B) = 4, n(B $\rm \cap$ C) = 9, n(C $\rm \cap$ A) = 11, and n ($\rm \overline { A \cup B \cup C }$) = 4, then

1. Find the value of n(U).
2. Find the value of n($\rm A \cup B \cup C$).
3. Show the information in a Venn diagram.
4. Is A a subset of (B $\rm \cup$ C)? Give reason.

If n(X) = 48, n(Y) = 51, n(Z) = 40, n(X $\rm \cap$ Y) = 11, n(Y $\rm \cap$ Z) = 10, n(Z $\rm \cap$ X) = 9, n(X $\rm \cap$ Y $\rm \cap$ Z) = 4 and $\rm n( \overline{X \cup Y \cup Z } )$ = 7, then

1. Find the value of $\rm n(X \cup Y \cup Z)$.
2. Find the value of n(U).
3. Show the information in a Venn diagram.
4. Find the cardinality of a set that is formed by the elements that are exactly in two of the sets X, Y, and Z. 

In a survey of 200 people, 120 like to play football, 85 like to play volleyball and 30 like to play none of these two games.

1. If F and V denote the sets of people who like to play football and volleyball respectively, then write the cardinality of $n(\overline {F \cup V})$.
2. Present the above information in a Venn diagram.
3. How many people like to play volleyball only? Find it.
4. Compare the number of people who like to play both the games and the number of people who don't like any of these two games.

A survey was conducted among 20 students of a Secondary School of class X, where 14 students participated in football and 12 students in volleyball games. Every student has participated in at least one game.

1. Assuming the set of students who participated in football as F and volleyball as V, write the cardinality of the set of students who participated in volleyball or football.
2. Present the above information in the Venn diagram.
3. Find the number of students participating in exactly one game by using Venn diagram.
4. If two students who respond only to football are unable to play the game due to health problems, the cardinality of which set is changed? Give reason.

If A and B are two non-empty sets with m and n (m > n) elements, respectively. What is the maximum number of elements in \( \rm (A \cap B) \)?

In a small village, two candidates A and B are competing against one another in an election. If each person voted without invalid votes, what can be the values of \( \rm n(A \cap B) \) and \( \rm n(A \cup B) \)?

Out of 90 students who participated in an examination, 43 passed in Science, 40 in Mathematics and 38 in Nepali. Among them, 13 passed in Science and Mathematics, 18 in Mathematics and Nepali as well as 16 passed in Science and Nepali. Using the information, answer the following questions: i) Show the information in Venn diagram. ii) Find the number of students who did not pass in any subject.

In 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. Based on the information, answer the following questions: i) Show the information in a Venn-diagram. ii) Find how many people were surveyed.

In a survey among 60 students, 23 played volleyball, 15 played basketball and 20 played cricket. If 7 played volleyball and basketball, 5 played basketball and cricket, 4 played volleyball and cricket but 15 played neither of the games. Based on this information, answer the following questions: i) Show the information in Venn-diagram. ii) Find how many students played all the three games. iii) How many played only volleyball and cricket.

In a survey of 125 students of class 10 in a school, 65 preferred to visit Lumbini, 75 preferred to visit Pokhara and 25 preferred both the places in their annual excursion.

(i) Show the above information in a Venn-diagram
(ii) Find the number of students who liked neither of two places.
(iii) Find the number of students who liked only Lumbini.

In a survey of 2000 Indian tourists who arrived in Nepal, 65% wished to visit Pashupati, 50% wished to visit Chandragiri and 45% wished to visit Manakamana. Similarly, 35% wished to visit Pashupati and Chandragiri, 25% to Chandragiri and Manakamana, and 20% to Manakamana and Pashupati. If 5% wished to visit none of these places, find the number of tourists who wished to visit all these three places.

In a survey of 300 people, it was found that 150 people like iPhones and 200 people like Android phones. But 25 people did not like any of these two phones.

1. If I and A denote the sets of people who like iPhone and Android Phones respectively, write the cardinality of $\rm n \overline {( I \cup A ) }$.
2. Present the above information in a Venn diagram.
3. Find the number of people who like iPhones only.
4. Compare the number of people who like both iPhone and Android phones and who do not like any of these two phones.

If n(A) = 35, n(B) = 30, and n(A $\rm \cap$ B) = 20, find n (A $\cup$ B) and n_o(A) by using a Venn diagram.

If n(X) = 40, n(Y) = 60 and n(X $\rm \cup$ Y) = 85, then find n(X $\rm \cap$ Y) and n_o(Y) by using formula.

If P = {Multiples of 2 up to 20}, Q = {Multiples of 3 up to 24} and U = {integers from 1 to 25}, find n_o(P) and n(P $\rm \cup$ Q) by using a Venn diagram.

If A = {factors of 18}, B = {multiples of 3 up to 27}, and U = {integers from 1 to 30}, find the cardinality of n_o(B) and n(A$\rm \cup$B) by using a Venn diagram.

In sets A and B, A has 40 members, B has 50 members, and $\rm (A \cup B)$ has 60 members. By how many elements $\rm (A \cap B)$ is formed?

In sets A and B, A has 50 members, B has 60 members, and 30 members are the same in both sets. By how many elements $\rm (A \cup B)$ is formed?

Out of 90 civil servants, 65 were working in the office, 50 were working in the field and 35 were working in both the premises (sites).

1. Let O and F represent the set of civil servants working in office and field respectively, then find $\rm n(F)$ and $\rm n(O \cap F)$.
2. Represent the given information in a Venn diagram.
3. How many civil servants were absent?
4. How many civil servants were working in the field only?

Out of 100 students, 80 passed in Science, 71 in Mathematics, 10 failed in both subjects, and 7 did not appear in an examination.

1. Let S and M represent the sets of students who were passed in science and Maths respectively, then find n(S) and n(M).
2. Find the number of students who passed in either Science or Mathematics.
3. Find the number of students who passed in both subjects.
4. Represent the above information in a Venn diagram.

In an examination, it was found that 55% failed in Maths and 45% failed in English. If 35% passed in both subjects

1. What percent failed in Maths only?
2. What percent failed in English only?
3. Represent the above information in a Venn diagram.
4. Find the percentage of students who failed in both subjects.

45% of the students of a school play basketball, 40% play cricket, and 30% play both. If 360 students play neither basketball nor cricket then answer the following questions:

1. If B and C represent the sets of students who like basketball and cricket, respectively, what percentage represents $\rm n \overline { \left ( C \cup B \right )} $ in the question?
2. Find the number of students who play either basketball or cricket.
3. Find the total number of students.
4. Find the number of students who play basketball only.

In an exam, 70% of the examinees passed in Science, 75% in Maths, 10% of them failed in both subjects, and 220 examinees passed in both subjects. If S and M are the set of examinees passed in Science and Maths respectively, then answer the following questions:

1. Find the percentage of n(S $\rm \cup$ M).
2. FInd the percentage of n(S $\rm \cap$ M).
3. Find the total number of students.
4. What percentage in the Venn diagram has been covered by n(S $\rm \triangle$ M)?

The population of the village is 15000. Among them 9000 read Magazine A, 7500 read Magazine B, and 40% read both magazines.

1. Find the percentage of $\rm n(A \cap B)$.
2. Find the value of $\rm n(A \cup B)$.
3. Find the number of people who don't read both magazines.
4. Find the percent of people who don't read both magazines.

In a class of 25 students, 12 have chosen Mathematics, 8 have chosen Mathematics but not Biology. If each of them has chosen at least one, then

1. What is the relation between the total number of students and the students who have not chosen Mathematics or Biology?
2. Find the number of students who have chosen both Mathematics and Biology.
3. Find the number of students who have chosen Mathematics but not Biology.
4. Find the cardinality of the set representing the symmetric difference of the set of the students who like Mathematics and Biology.

In a class of 65 students, 10 students liked Maths but not English, and 20 students liked English but not Maths. If 5 students did not like both then,

1. Find the number of students who liked Maths or Science.
2. Find the number of students who liked both Maths and Science.
3. Show the given information in a Venn-diagram.
4. Find the cardinality of the set representing the symmetric difference of the set of students who like Maths and Science.

Out of 100 students of class V, 73 passed in Mathematics and 84 in Nepali in the final examination but 7 failed in both subjects and 5 were absent in the examination.

1. If M and N represent the set of students who passed in Mathematics and Nepali then what are the values of n(M) and n(N)?
2. Find the total number of students who failed in both subjects.
3. Find the number of students who passed in either of the subjects.
4. Find the number of students who passed in both subjects.
5. Show the given information in a Venn-diagram. Which region in the Venn-diagram represents the maximum number of students?

Out of 75 students of class X, 30 passed in Mathematics and 40 in Social Studies in the final examination, but 10 failed in both subjects and 5 were absent in the examination.

1. If M and S represent the set of students who passed in Maths and Social Studies, find the value of n(M) and n(S).
2. Find the total number of students who failed in both subjects.
3. Find the number of students who passed in both subjects.
4. Show the given information in a Venn-diagram. Which region in the Venn-diagram represents the minimum number of students?

Out of 100 students in a class, 20 students like Maths but not Science, and 30 students like Science but not Maths. If 20 students like neither of the subjects

1. Let M and S represent the set of students who like Maths and Science respectively then write the value of $\rm n_o(M)$ and $\rm n_o(S)$.
2. Show the given information in a Venn-diagram.
3. Find the number of students who like both Maths and Science.
4. Find the ratio of the students who like Maths to the students who like Science. 

250 students in a group were asked whether they like mango or apple. 80 students like mango but not apple and 50 students like apple but not mango. If 50 students do not like both of the fruits, then,

1. Let M and A represent the set of students who like mangoes and apples respectively then write the value of $\rm n_o(M)$ and $\rm n_o(A)$.
2. Show the given information in a Venn diagram.
3. Find the number of students who like both mangoes and apples.
4. Find the ratio of the students who like mango to the students who like apple.

1000 students in a school like Nepali, History, or both subjects. Out of them, 400 like both subjects. If the ratio of the number of students who like Nepali and History is 3:2, by using the Venn diagram, find

1. Find the number of students who like Nepali.
2. Find the number of students who like History.
3. Find the number of students who like only one subject.
4. Show the results in a Venn diagram.

In a school, 640 teachers like either milk or curd or both. The ratio of the number of teachers who like milk to the number of teachers who like curd is 3:2 and 160 teachers like both milk and curd. Find:

1. Find the number of students who like milk.
2. Find the number of students who like curd.
3. Find the number of students who like only one: milk or curd.
4. Show the results in a Venn diagram.

If n(A) = 40, n(B) = 60, and n(A $\rm \cup$ B) = 80,

1. Find the value of n (A $\rm \cap$ B).
2. Draw a Venn diagram of the above information.

A and B are two subsets of a universal set U in which n(U) = 43, n(A) = 25, n(B) = 18, and n(A $\rm \cap$ B) = 7.

1. Draw a Venn diagram of the above information.
2. Find the value of $\rm n \overline{ A \cup B} $.

If A and B are two subsets of a universal set U in which, n(U) = 70, n(A) = 40, n(B) = 20, and $\rm n \overline { A \cup B}$ = 15 then

1. Show the above information in a Venn diagram.
2. Find the value of $\rm n ( A \cap B)$.

If n(A) = 40, n(B) = 60 and $\rm n (A \cup B) = 80$, then

1. find the value of $\rm n( A \cap B)$.
2. find the value of $\rm n_o (A)$.
3. show it in a Venn Diagram.

In a survey of 60 students, 30 drink milk, 25 drink curd, and 10 students drink milk as well as curd.

1. If the set of total students is U, then write the cardinality of U.
2. Draw a Venn diagram to illustrate the above information.
3. Find the number of students who drink either of them.
4. Find the number of students who drink neither of them.

A survey of a community shows that 55% of the people like to listen to the radio, 65% like to watch the television, and 35% like to listen radio as well as to watch the television.

1. If the set of people in the community is U then write the cardinality of U in percentage.
2. Show the above information in a Venn-diagram.
3. Find the percentage of people who like either to listen to the radio or to watch the television.
4. Find the percentage of people who do not like to listen to the radio as well as to watch the television.

In a class of 30 students, 20 students like to play cricket and 15 like to play volleyball. Also, each student likes to play at least one of the two games.

1. If C and V be the set of students who like cricket and volleyball respectively then find the cardinality of C and V.
2. Illustrate the above information in a Venn diagram.
3. Find the number of students who like either of the games.
4. How many students like to play both the games?

In a survey before an election, 65% of people liked leader A and 60% of people liked leader B. If 15% of people did not like to open their opinion about any of the leaders,

1. If A and B denote the set of people who like leader A and leader B respectively and n(U) = 100 then find n(A) and n(B).
2. Show the given information in a Venn-diagram.
3. Find the percentage of people who liked both the leaders by using the Venn diagram.
4. Find the percentage of people who like either of the leaders.

In a group of 100 students, 68 liked the football game, 60 liked the volleyball game and 2 didn't like any of the games.

1. Shwo the given information in a Venn diagram.
2. How many students like both games?
3. How many students like only football?
4. How many students like either of the games?

In a survey of 119 students, it was found that 16 drink neither milk nor tea and 69 drink milk and tea.

1. Show the given information in a Venn diagram.
2. How many students drink milk only?
3. How many students drink tea only?
4. How many students like either of the drinks?

If A = {a,c,e}, B = {b,c,d}, and C = {a,c,d,f}, find $\rm n(A \cap B \cap C)$.

If A = {1, 2, 3}, B = {2, 3, 4}, and C = {3, 4, 5}, find $\rm n(A \cup B \cup C)$.

If n(A) = 65, n(B) = 50, n(C) = 35, n(A $\rm \cap$ B) = 25, n(B $\rm \cap$ C) = 20, n(C $\rm \cap$ A) = 15, n(A $\rm \cap$ B $\rm \cap$ C) = 5, and n(U) = 100,

1. Are the sets A, B, and C overlapping sets? Give reason.
2. Find the value of $\rm n(A \cup B \cup C)$.
3. Find the value of $\rm n \overline{ A \cup B \cup C }$.
4. Show the given information in a Venn diagram.

If n(A) = 48, n(B) = 51, n(C) = 40, n(A $\rm \cap$ B) = 11, n(B $\rm \cap$ C) = 10, n(C $\rm \cap$ A) = 9, n(A $\rm \cap$ B $\rm \cap$ C) = 4 and n(U) = 120,

1. Are the sets A, B, and C overlapping sets? Give reason.
2. Find the value of n($\rm A \cup B\cup C$).
3. Find the value of n($\rm \overline{A \cup B \cup C}$).
4. Present the above information in a Venn diagram.

Sets A, B, and C are the subsets of the universal set U. If n(U) = 300, n(A) = 100, n(B) = 90, n(C) = 110, n(A$\rm \cap$B) = 60, n(B$\rm \cap$C) = 40, n(C$\rm \cap$A) = 45, and n(A $\rm \cup$ B $\rm \cup$ C) = 200.

1. The sets A, B, and C are overlapping sets. Give reason.
2. Find the value of n(A $\rm \cap$ B $\rm \cap$ C).
3. Find the value of n($\rm \overline{A \cup B \cup C}$).
4. Present the given information in a Venn diagram.

A, B and C are the subsets of the universal set U. If n(U) = 100, n(A) = 60, n(B) = 45, n(C) = 30, n(A $\rm \cap$ B) = 25, n(B $\rm \cap$ C) = 20, n(C $\rm \cap$ A) = 15, n(A $\rm \cup$ B $\rm \cup$ C) = 85.

1. The sets A, B, and C are overlapping sets. Give reason.
2. Find the value of n(A $\rm \cap$ B $\rm \cap$ C).
3. Find the value of n($\rm \overline{A \cup B \cup C}$).
4. Present the given information in a Venn diagram.

If n(A) = 12, n(B) = 12, n(A $\rm \cap$ B) = 5, n(A $\rm \cap$ C) = 3, n(B $\rm \cap$ C) = 4, n(A $\rm \cap$ B $\rm \cap$ C) = 2 and n(A $\rm \cup$ B $\rm \cup$ C) = 20, then

1. Find the value of $\rm n_{o}(C)$.
2. Find the value of $\rm n(C)$.
3. Find the value of $\rm n(A \cup C)$.
4. Present in a Venn diagram.

If n(A) = 36, n(B) = 36, n(A $\rm \cap$ B) = 15, n(A $\rm \cap$ C) = 15, n(B $\rm \cap$ C) = 12, n(A $\rm \cup$ B $\rm \cup$ C) = 66, then

1. Find the value of $\rm n_{o}(C)$.
2. Find the value of $\rm n(C)$.
3. Find the value of $\rm n(A \cup C)$.
4. Present in a Venn diagram.

If n(A) = 14, n(B) = 13, n(C) = 22, n(A $\rm \cap$ B $\rm \cap$ C) = 4, n(A $\rm \cap$ B) = 4, n(B $\rm \cap$ C) = 9, n(C $\rm \cap$ A) = 11, and n ($\rm \overline { A \cup B \cup C }$) = 4, then

1. Find the value of n(U).
2. Find the value of n($\rm A \cup B \cup C$).
3. Show the information in a Venn diagram.
4. Is A a subset of (B $\rm \cup$ C)? Give reason.

If n(X) = 48, n(Y) = 51, n(Z) = 40, n(X $\rm \cap$ Y) = 11, n(Y $\rm \cap$ Z) = 10, n(Z $\rm \cap$ X) = 9, n(X $\rm \cap$ Y $\rm \cap$ Z) = 4 and $\rm n( \overline{X \cup Y \cup Z } )$ = 7, then

1. Find the value of $\rm n(X \cup Y \cup Z)$.
2. Find the value of n(U).
3. Show the information in a Venn diagram.
4. Find the cardinality of a set that is formed by the elements that are exactly in two of the sets X, Y, and Z. 

In a survey of 200 people, 120 like to play football, 85 like to play volleyball and 30 like to play none of these two games.

1. If F and V denote the sets of people who like to play football and volleyball respectively, then write the cardinality of $n(\overline {F \cup V})$.
2. Present the above information in a Venn diagram.
3. How many people like to play volleyball only? Find it.
4. Compare the number of people who like to play both the games and the number of people who don't like any of these two games.

A survey was conducted among 20 students of a Secondary School of class X, where 14 students participated in football and 12 students in volleyball games. Every student has participated in at least one game.

1. Assuming the set of students who participated in football as F and volleyball as V, write the cardinality of the set of students who participated in volleyball or football.
2. Present the above information in the Venn diagram.
3. Find the number of students participating in exactly one game by using Venn diagram.
4. If two students who respond only to football are unable to play the game due to health problems, the cardinality of which set is changed? Give reason.

If A and B are two non-empty sets with m and n (m > n) elements, respectively. What is the maximum number of elements in \( \rm (A \cap B) \)?

In a small village, two candidates A and B are competing against one another in an election. If each person voted without invalid votes, what can be the values of \( \rm n(A \cap B) \) and \( \rm n(A \cup B) \)?