Set Theory 1

1. Let $A=\{(1,2),(3,4), 5\}$, then which of the following is incorrect?

2. A market research group conducted a survey of 1000 consumers and reported that 720 consumers liked product A and 450 consumers liked product $\mathrm{B}$. What is the least number that must have liked both products ?

3.One of the partitions of the set $\{1,2,5, x, y, \sqrt{2}, \sqrt{3}\}$ is

4. Let $\mathrm{A}$ and $\mathrm{B}$ be two sets then $(\mathrm{A} \cup \mathrm{B})^{\prime} \cup\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)$ is equal to

5. Let $\mathrm{A}=\{(\mathrm{n}, 2 \mathrm{n}): \mathrm{n} \in \mathrm{N}\}$ and $\mathrm{B}=\{(2 \mathrm{n}, 3 \mathrm{n}): \mathrm{n} \in \mathrm{N}\}$. What is $\mathrm{A} \cap \mathrm{B}$ equal to ?

6. If $a \mathrm{~N}=\{\mathrm{ax}: \mathrm{x} \in \mathbf{N}\}$ and $\mathrm{bN} \cap \mathrm{cN}=\mathrm{dN}$, where $\mathrm{b}, \mathrm{c} \in \mathbf{N}$ are relatively prime, then

7. In a class of 55 students, the number of students studying different subjects are 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is

8. A set A has 3 elements and another set B has 6 elements.Then

9. If $\mathrm{A}=\{1,2,5\}$ and $\mathrm{B}=\{3,4,5,9\}$, then $\mathrm{A} \Delta \mathrm{B}$ is equal to

10. At a certain conference of 100 people, there are 29 Indian women and 23 Indian men. Of these Indian people 4 are doctors and 24 are either men or doctors. There are no foreign doctors. How many foreigners and women doctors are attending the conference?

11. Let $X$ and $Y$ be two non-empty sets such that $X \cap A=Y \cap A=\phi$ and $X \cup A=Y \cup A$ for some non-empty set $A$. Then

12. Let $A$ and $B$ are two sets in a universal set $U$. Then which of these is/are correct?

13. If $A$ and $B$ are non-empty sets such that $A \supset B$, then

14. In a town of 10,000 families, it was found that $40 \%$ families buy newspaper A, $20 \%$ families buy newspaper B and $10 \%$ families buy newspaper C. $5 \%$ families buy $\mathrm{A}$ and B, $3 \%$ buy $B$ and $C$ and $4 \%$ buy $A$ and $C$. If $2 \%$ families buy all the newspapers, then

15. In a battle $70 \%$ of the combatants lost one eye, $80 \%$ an ear, $75 \%$ an arm, $85 \%$ a leg, $x \%$ lost all the four limbs. The minimum value of $x$ is

16. Let $n(\mathrm{U})=700, n(\mathrm{~A})=200, n(\mathrm{~B})=300, n(\mathrm{~A} \cap \mathrm{B})=100$, then $n\left(\mathrm{~A}^{\prime} \cap \mathrm{B}^{\prime}\right)$ is equal to

17. Statement-1 : If $\mathrm{B}=\mathrm{U}-\mathrm{A}$, then $n(\mathrm{~B})=n(\mathrm{U})-n(\mathrm{~A})$ where $\mathrm{U}$ is universal set. Statement-2 : For any three arbitrary set A, B, C we have if $\mathrm{C}=\mathrm{A}-\mathrm{B}$, then $n(\mathrm{C})=n(\mathrm{~A})-n(\mathrm{~B})$.

18. Each student in a class of 40 , studies at least one of the subjects English, Mathematics and Economics. 16 study English, 22 Economics and 26 Mathematics, 5 study English and Economics, 14 Mathematics and Economics and 2 study all the three subjects. The number of students who study English and Mathematics but not Economics is

19. In a class of 80 students numbered a to 80 , all odd numbered students opt of Cricket, students whose numbers are divisible by 5 opt for Football and those whose numbers are divisible by 7 opt for Hockey. The number of students who do not opt any of the three games, is

20. In a class of 60 students, 23 play Hockey 15 Play Basket-ball and 20 play cricket. 7 play Hockey and Basket-ball, 5 play cricket and Basket-ball, 4 play Hockey and Cricket and 15 students do not play any of these games. Then

21. The set $(A \backslash B) \cup(B \backslash A)$ is equal to

22. If $\mathrm{A}$ is the set of the divisors of the number $15, \mathrm{~B}$ is the set of prime numbers smaller than 10 and $\mathrm{C}$ is the set of even numbers smaller than 9 , then $(\mathrm{A} \cup \mathrm{C}) \cap \mathrm{B}$ is the set

23. Two finite sets have $m$ and $n$ elements. The number of subsets of the first set is 112 more than that of the second set. The values of $m$ and $n$ are, respectively,

24. The number of students who take both the subjects mathematics and chemistry is 30 . This represents $10 \%$ of the enrolment in mathematics and $12 \%$ of the enrolment in chemistry. How many students take at least one of these two subjects?

25. If $n(\mathrm{~A})=1000, n(\mathrm{~B})=500$ and if $n(\mathrm{~A} \cap \mathrm{B}) \geq 1$ and $n(\mathrm{~A} \cup \mathrm{B})=p$, then

26. The number of elements in the set $\left\{(a, b): 2 a^{2}+3 b^{2}=35, a, b \in \mathrm{Z}\right\}$, where $\mathrm{Z}$ is the set of all integers, is

27. Let A, B, C be finite sets. Suppose that $n(A)=10, n(B)=15, n$ $(C)=20, n(A \cap B)=8$ and $n(B \cap C)=9$. Then the possible value of $n(A \cup B \cup C)$ is

28. The value of $(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}) \cap\left(\mathrm{A} \cap \mathrm{B}^{c} \cap \mathrm{C}^{c}\right)^{c} \cap \mathrm{C}^{c}$, is

29. In a town of 10,000 families it was found that $40 \%$ family buy newspaper A, $20 \%$ buy newspaper B and $10 \%$ families buy newspaper C, $5 \%$ families buy $\mathrm{A}$ and $\mathrm{B}, 3 \%$ buy $\mathrm{B}$ and $\mathrm{C}$ and $4 \%$ buy $\mathrm{A}$ and $\mathrm{C}$. If $2 \%$ families buy all the three newspapers, then number of families which buy A only is

30. Statement-1 : If $\mathrm{A} \cup \mathrm{B}=\mathrm{A} \cup \mathrm{C}$ and $\mathrm{A} \cap \mathrm{B}=\mathrm{A} \cap \mathrm{C}$, then $\mathrm{B}=\mathrm{C}$. Statement-2 $: \mathrm{A} \cup(\mathrm{B} \cap \mathrm{C})=(\mathrm{A} \cup \mathrm{B}) \cap(\mathrm{A} \cup \mathrm{C})$.