Sets and Relations

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 elements are

Let $\mathop \cup \limits_{i = 1}^{50} {X_i} = \mathop \cup \limits_{i = 1}^n {Y_i} = T$ where each X_i contains 10 elements and each Y_i contains 5 elements. If each element of the set T is an element of exactly 20 of sets X_i’s and exactly 6 of sets Y_i’s, then n is equal to :

The number of elements in the set {x $\in$ R : (|x| $-$ 3) |x + 4| = 6} is equal to :

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 $\cap$ Z is empty, is :

Let P = {$\theta$ : sin$\theta$ $-$ cos$\theta$ = $\sqrt 2 \,\cos \theta$} and Q = {$\theta$ : sin$\theta$ + cos$\theta$ = $\sqrt 2 \,\sin \theta$} be two sets. Then

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 $\times$ R : 4($a$ - 6)² + 9(b - 5)² $\le$ 36 }; Then

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 :

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}, then the number of subsets of the set A $\times$ B, is

If R = {(x, y) : x, y $\in$ Z, x² + 3y² $\le$ 8} is a relation on the set of integers Z, then the domain of R^–1 is :

Consider the two sets : A = {m $\in$ R : both the roots of x² – (m + 1)x + m + 4 = 0 are real} and B = [–3, 5). Which of the following is not true?

Let A = {2, 3, 4, 5, ....., 30} and '$\simeq$' be an equivalence relation on A $\times$ A, defined by (a, b) $\simeq$ (c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :

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}. Then, the number of elements in R is :

Let R be the real line. Consider the following subsets of the plane $R \times R$ : $S = \left\{ {(x,y):y = x + 1\,\,and\,\,0 < x < 2} \right\}$ $T = \left\{ {(x,y): x - y\,\,\,is\,\,an\,\,{\mathop{\rm int}} eger\,} \right\}$, Which one of the following is true ?

Let $\mathrm{A}=\{1,3,4,6,9\}$ and $\mathrm{B}=\{2,4,5,8,10\}$. Let $\mathrm{R}$ be a relation defined on $\mathrm{A} \times \mathrm{B}$ such that $\mathrm{R}=\left\{\left(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\right): a_{1} \leq b_{2}\right.$ and $\left.b_{1} \leq a_{2}\right\}$. Then the number of elements in the set R is :

Let $\mathrm{A}=\{2,3,4\}$ and $\mathrm{B}=\{8,9,12\}$. Then the number of elements in the relation $\mathrm{R}=\left\{\left(\left(a_{1}, \mathrm{~b}_{1}\right),\left(a_{2}, \mathrm{~b}_{2}\right)\right) \in(A \times B, A \times B): a_{1}\right.$ divides $\mathrm{b}_{2}$ and $\mathrm{a}_{2}$ divides $\left.\mathrm{b}_{1}\right\}$ is :

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $P(m, n)$ from the point $Q(-2,-3)$ is :

If R is the smallest equivalence relation on the set $\{1,2,3,4\}$ such that $\{(1,2),(1,3)\} \subset \mathrm{R}$, then the number of elements in $\mathrm{R}$ is __________.

Let $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ be a relation on the set $A=\{1,2,3,4\}$. The relation $R$ is :

Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is :