Write the following sets in the roaster form.

(i) $A=\{x: x \in R, 2 x+11=15\}$

(ii) $B=\left\{x \mid x^2=x, x \in R\right\}$

(iii) $C=\{x \mid x$ is a positive factor of a prime number $p\}$

Write the following sets in the roaster form.

(i) $D=\left\{t \mid t^3=t, t \in R\right\}$

(ii) $E=\left\{w \left\lvert\, \frac{w-2}{w+3}=3\right., w \in R\right\}$

(iii) $F=\left\{x \mid x^4-5 x^2+6=0, x \in R\right\}$

If $Y=\left\{x \mid x\right.$ is a positive factor of the number $2^{p-1}\left(2^p-1\right)$, where $2^p-1$ is a prime number\}. Write $Y$ in the roaster form.

State which of the following statements are true and which are false. Justify your answer.

(i) $35 \in\{x \mid x$ has exactly four positive factors $\}$.

(ii) $128 \in\{y \mid$ the sum of all the positive factors of $y$ is $2 y\}$.

(iii) $3 \notin\left\{x \mid x^4-5 x^3+2 x^2-112 x+6=0\right\}$.

(iv) $496 \notin\{y \mid$ the sum of all the positive factors of $y$ is $2 y\}$.

If $L=\{1,2,3,4\}, M=\{3,4,5,6\}$ and $N=\{1,3,5\}$, then verify that $L-(M \cup N)=(L-M) \cap(L-N)$.

If $A$ and $B$ are subsets of the universal set $U$, then show that

(i) $A \subset A \cup B$

(ii) $A \subset B \Leftrightarrow A \cup B=B$

(iii) $(A \cap B) \subset A$

Given that $N=\{1,2,3, \ldots, 100\}$. Then, write

(i) the subset of $N$ whose elements are even numbers.

(ii) the subset of $N$ whose elements are perfect square numbers.

If $X=\{1,2,3\}$, if $n$ represents any member of $X$, write the following sets containing all numbers represented by

(i) $4 n$

(ii) $n+6$

(iii) $\frac{n}{2}$

(iv) $n-1$

If $Y=\{1,2,3, \ldots, 10\}$ and $a$ represents any element of $Y$, write the following sets, containing all the elements satisfying the given conditions.

(i) $a \in Y$ but $a^2 \notin Y$

(ii) $a+1=6, a \in Y$

(iii) $a$ is less than 6 and $a \in Y$

$A, B$ and $C$ are subsets of universal set $U$. If $A=\{2,4,6,8,12,20\}$, $B=\{3,6,9,12,15\}, C=\{5,10,15,20\}$ and $U$ is the set of all whole numbers, draw a Venn diagram showing the relation of $U, A, B$ and $C$.

Let $U$ be the set of all boys and girls in a school, $G$ be the set of all girls in the school, $B$ be the set of all boys in the school and $S$ be the set of all students in the school who take swimming. Some but not all, students in the school take swimming. Draw a Venn diagram showing one of the possible interrelationship among sets $U, G, B$ and $S$.

For all sets $A, B$ and $C$, show that $(A-B) \cap(A-C)=A-(B \cup C)$.

For all sets $A$ and $B,(A-B) \cup(A \cap B)=A$.

For all sets $A, B$ and $C, A-(B-C)=(A-B)-C$.

For all sets $A, B$ and $C$, if $A \subset B$, then $A \cap C \subset B \cap C$.

For all sets $A, B$ and $C$, if $A \subset B$, then $A \cup C \subset B \cup C$.

For all sets $A, B$ and $C$, if $A \subset C$ and $B \subset C$, then $A \cup B \subset C$.

For all sets $A, B$ and $C$, if $A \subset C$ and $B \subset C$, then $A \cup B \subset C$.

For all sets $A$ and $B, A \cup(B-A)=A \cup B$.

For all sets $A$ and $B, A-(A-B)=A \cap B$.

For all sets $A$ and $B, A-(A \cap B)=A-B$.

For all sets $A$ and $B,(A \cup B)-B=A-B$.

Let $T=\left\{x \left\lvert\, \frac{x+5}{x-7}-5=\frac{4 x-40}{13-x}\right.\right\}$. Is $T$ an empty set? Justify your answer.

If $A, B$ and $C$ be sets. Then, show that $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$.

Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science, 4 in English and Science, 4 in all the three. Find how many passed

(i) in English and Mathematics but not in Science.

(ii) in Mathematics and Science but not in English.

(iii) in Mathematics only.

(iv) in more than one subject only.

In a class of 60 students, 25 students play cricket and 20 students play tennis and 10 students play both the games. Find the number of students who play neither.

In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics and 70 study Chemistry, 40 study Mathematics and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Mathematics and 20 none of these subjects. Find the number of students who study all the three subjects.

In a town of 10000 families, it was found that $40 \%$ families buy newspaper $A, 20 \%$ families buy newspaper $B, 10 \%$ families buy newspaper $C, 5 \%$ families buy $A$ and $B, 3 \%$ buy $B$ and $C$ and $4 \%$ buy $A$ and $C$. If $2 \%$ families buy all the three newspaper. Find

(i) the number of families which buy newspaper $A$ only.

(ii) the number of families which buy none of $A, B$ and $C$.

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows French =17, English = 13, Sanskrit $=15$ French and English = 09, English and Sanskrit $=4$, French and Sanskrit = 5, English, French and Sanskrit $=3$. Find the number of students who study

(i) only French.

(ii) only English.

(iii) only Sanskrit.

(iv) English and Sanskrit but not French.

(v) French and Sanskrit but not English.

(vi) French and English but not Sanskrit.

(vii) atleast one of the three languages.

(viii) none of the three languages.

Suppose, $A_1, A_2, \ldots, A_{30}$ are thirty sets each having 5 elements and $B_1, B_2, B_n$ are $n$ sets each with 3 elements, let $\bigcup_\limits{i=1}^{30} A_i=\bigcup_\limits{j=1}^n B_j=S$ and each element of $S$ belongs to exactly 10 of the $A_i{ }^{\prime}$ 's and exactly 9 of the $B_j$ 's. Then, $n$ is equal to

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

The set $\left(A \cap B^{\prime}\right)^{\prime} \cup(B \cap C)$ is equal to

Let $F_1$ be the set of parallelograms, $F_2$ the set of rectangles, $F_3$ the set of rhombuses, $F_4$ the set of squares and $F_5$ the set of trapeziums in a plane. Then, $F_1$ may be equal to

Let $S=$ set of points inside the square, $T=$ set of points inside the triangle and $C=$ set of points inside the circle. If the triangle and circle intersect each other and are contained in a square. Then,

If $R$ be the set of points inside a rectangle of sides $a$ and $b(a, b>1)$ with two sides along the positive direction of $X$-axis and $Y$-axis. Then,

In a town of 840 persons, 450 persons read Hindi, 300 read English and 200 read both. Then, the number of persons who read neither, is

If $X=\left\{8^n-7 n-1 \mid n \in N\right\}$ and $y=\{49 n-49 \mid n \in N\}$. Then,

A survey shows that $63 \%$ of the people watch a news channel whereas $76 \%$ watch another channel. If $x \%$ of the people watch both channel, then

If sets $A$ and $B$ are defined as $A=\left\{(x, y) \left\lvert\, y=\frac{1}{x}\right., 0 \neq x \in R\right\}, B=\{(x, y) \mid y=-x, x \in R$,$\} . Then,$

If $A$ and $B$ are two sets, then $A \cap(A \cup B)$ equals to

If $A=\{1,3,5,7,9,11,13,15,17\}, B=\{2,4, \ldots, 18\}$ and $N$ the set of natural numbers is the universal set, then $\left(A^{\prime} \cup(A \cup B) \cap B^{\prime}\right)$ is

If $S=\{x \mid x$ is a positive multiple of 3 less than 100 $\}$ and $P=\{x \mid x$ is a prime number less than 20$\}$. Then, $n(S)+n(P)$ is equal to

If $X$ and $Y$ are two sets and $X^{\prime}$ denotes the complement of $X$, then $X \cap(X \cup Y)^{\prime}$ is equal to

The set $\{x \in R: 1 \leq x<2\}$ can be written as ............. .

When $A=\phi$, then number of elements in $P(A)$ is .............. .

If $A$ and $B$ are finite sets, such that $A \subset B$, then $n(A \cup B)$ is equal to ............ .

If $A$ and $B$ are any two sets, then $A-B$ is equal to ............. .

Power set of the set $A=\{1,2\}$ is ........... .

If the sets $A=\{1,3,5\}, B=\{2,4,6\}$ and $C=\{0,2,4,6,8\}$. Then, the universal set of all the three sets $A, B$ and $C$ can be .......... .

If $U=\{1,2,3,4,5,6,7,8,9,10\}, A=\{1,2,3,5\}, B=\{2,4,6,7\}$ and $C=\{2,3,4,8\}$. Then,

(i) $(B \cup C)^{\prime}$ is ............ .

(ii) $(C-A)^{\prime}$ is .......... .

For all sets $A$ and $B, A-(A \cap B)$ is equal to .......... .

Match the following sets for all sets $A, B$ and $C$

If $A$ is any set, then $A \subset A$.

If $M=\{1,2,3,4,5,6,7,8,9\}$ and $B=\{1,2,3,4,5,6,7,8,9\}$, then $B \not \subset M$.

The sets $\{1,2,3,4\}$ and $\{3,4,5,6\}$ are equal

$Q \cup Z=Q$, where $Q$ is the set of rational numbers and $Z$ is the set of integers.

Let sets $R$ and $T$ be defined as $$ \begin{aligned} & R=\{x \in Z \mid x \text { is divisible by } 2\} \\ & T=\{x \in Z \mid x \text { is divisible by } 6\} . \text { Then, } T \subset R \end{aligned}$$

Given $A=\{0,1,2\}, B=\{x \in R \mid 0 \leq x \leq 2\}$. Then, $A=B$