Let $$P(S)$$ denote the power set of $$S=\{1,2,3, \ldots ., 10\}$$. Define the relations $$R_{1}$$ and $$R_{2}$$ on $$P(S)$$ as $$\mathrm{AR}_{1} \mat...

Let $$R$$ be a relation on $$1\mathbb{R}$$, given by $$R=\{(a, b): 3 a-3 b+\sqrt{7}$$ is an irrational number $$\}$$. Then $$R$$ is

Among the relations $\mathrm{S}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R}-\{0\}, 2+\frac{\mathrm{a}}{\mathrm{b}}>0\right...

Let $$\mathrm{R}$$ be a relation on $$\mathrm{N} \times \mathbb{N}$$ defined by $$(a, b) ~\mathrm{R}~(c, d)$$ if and only if $$a d(b-c)=b c(a-d)$$. Th...

The minimum number of elements that must be added to the relation $$\quad \mathrm{R}=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{c})\}$$ on the se...

Let R be a relation defined on $$\mathbb{N}$$ as $$a\mathrm{R}b$$ if $$2a+3b$$ is a multiple of $$5,a,b\in \mathbb{N}$$. Then R is

The relation $$\mathrm{R = \{ (a,b):\gcd (a,b) = 1,2a \ne b,a,b \in \mathbb{Z}\}}$$ is :

Let R be a relation from the set $$\{1,2,3, \ldots, 60\}$$ to itself such that $$R=\{(a, b): b=p q$$, where $$p, q \geqslant 3$$ are prime numbers}. T...

$$ \text { Let } S=\left\{x \in[-6,3]-\{-2,2\}: \frac{|x+3|-1}{|x|-2} \geq 0\right\} \text { and } $$$$T=\left\{x \in \mathbb{Z}: x^{2}-7|x|+9 \leq 0\...

For $$\alpha \in \mathbf{N}$$, consider a relation $$\mathrm{R}$$ on $$\mathbf{N}$$ given by $$\mathrm{R}=\{(x, y): 3 x+\alpha y$$ is a multiple of 7$...

Let $$R_{1}$$ and $$R_{2}$$ be two relations defined on $$\mathbb{R}$$ by $$a \,R_{1} \,b \Leftrightarrow a b \geq 0$$ and $$a \,R_{2} \,b \Leftrighta...

Let a set A = A1 $$\cup$$ A2 $$\cup$$ ..... $$\cup$$ Ak, where Ai $$\cap$$ Aj = $$\phi$$ for i $$\ne$$ j, 1 $$\le$$ j, j $$\le$$ k. Define the relatio...

Let R1 = {(a, b) $$\in$$ N $$\times$$ N : |a $$-$$ b| $$\le$$ 13} and R2 = {(a, b) $$\in$$ N $$\times$$ N : |a $$-$$ b| $$\ne$$ 13}. Then on N :...

Let $$A = \{ x \in R:|x + 1|

Let A, B and C be sets such that $$\phi $$ $$ \ne $$ A $$ \cap $$ B $$ \subseteq $$ C. Then which of the following statements is not true ?

Two newspapers A and B are published in a city. It is known that 25% of the city populations reads A and 20% reads B while 8% reads both A and B. Furt...

Let Z be the set of integers. If A = {x $$ \in $$ Z : 2(x + 2) (x2 $$-$$ 5x + 6) = 1} and B = {x $$ \in $$ Z : $$-$$ 3 < 2x $$-$$ 1 < 9}, t...

Let S = {1, 2, 3, … , 100}. The number of non-empty subsets A of S such that the product of elements in A is even is :

The number of functions f from {1, 2, 3, ...., 20} onto {1, 2, 3, ...., 20} such that f(k) is a multiple of 3, whenever k is a multiple of 4, is :

In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics ...

Two sets A and B are as under : A = {($$a$$, b) $$ \in $$ R $$ \times $$ R : |$$a$$ - 5| < 1 and |b - 5| < 1}; B = {($$a$$, b) $$ \in $$ R $$ \t...

Let A and B be two sets containing four and two elements respectively. Then, the number of subsets of the set A $\times$ B , each having atleast three...

Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y $$ \subseteq $$ X, Z $$ \subseteq $$ X and Y $$ \...

Let S = {1, 2, 3, 5, 7, 10, 11}. The smaller of non-empty subsets of S that have the sum of all elements a multiple of 3, is _____________.

The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence re...

Let $$S=\{4,6,9\}$$ and $$T=\{9,10,11, \ldots, 1000\}$$. If $$A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\...

Let $$A=\{1,2,3,4,5,6,7\}$$ and $$B=\{3,6,7,9\}$$. Then the number of elements in the set $$\{C \subseteq A: C \cap B \neq \phi\}$$ is ___________....

Let $$A=\{1,2,3,4,5,6,7\}$$. Define $$B=\{T \subseteq A$$ : either $$1 \notin T$$ or $$2 \in T\}$$ and $$C=\{T \subseteq A: T$$ the sum of all the ele...

Let S = {1, 2, 3, 4}. Then the number of elements in the set { f : S $$\times$$ S $$\to$$ S : f is onto and f (a, b) = f (b, a) $$\ge$$ a $$\forall$$ ...

Let R1 and R2 be relations on the set {1, 2, ......., 50} such that R1 = {(p, pn) : p is a prime and n $$\ge$$ 0 is an integer} and R2 = {(p, pn) : p ...

Let A = {1, a1, a2 ....... a18, 77} be a set of integers with 1 1 2 18 1 + a2 + ...... + a18 is equal to _____________....

Let A = {n $$\in$$ N : H.C.F. (n, 45) = 1} and Let B = {2k : k $$\in$$ {1, 2, ......., 100}}. Then the sum of all the elements of A $$\cap$$ B is ____...

The sum of all the elements of the set $$\{ \alpha \in \{ 1,2,.....,100\} :HCF(\alpha ,24) = 1\} $$ is __________.

Let  A = {n$$ \in $$N: n is a 3-digit number}        B = {9k + 2: k $$ \in $$ N} and C = {9k + $$l$$: k $$ \in...