Sets and Relations

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 R1
and R2
be two relation defined as
follows :
R1
= {(a, b) \(\in\) R2
: a2
+ b2 \(\in\) Q} and
R2
= {(a, b) \(\in\) R2
: a2
+ b2 \(\notin\) Q},
where Q is the
set of all rational numbers. Then :

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 :

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. Further, 30% of those who read A but not B
look into advertisements and 40% of those who
read ...

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

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 ?

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 course and those whose number is divisible by 5 opted Chemistry course. Then the number of...

Let N denote the set of all natural numbers. Define two binary relations on N as R = {(x, y) \(\in\) N \(\times\) N : 2x + y = 10} and R2 = {(x, y) \(\in\) N \(\times\) N : x + 2y = 10}. Then :

Consider the following two binary relations on the set A = {a, b, c} :
R1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and
R2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}.
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 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

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 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 $R$ be the set of real numbers.
Statement I : $A=\{(x, y) \in R \times R: y-x$ is an integer $\}$ is an equivalence relation on $R$.
Statement II : $ B=\{(x, y) \in R \times R: x=\alpha y$ for some rational number $\alpha\}$ is an equiv...

Consider the following relations
$R=\{(x, y) \mid x, y$ are real numbers and $x=w y$ for some rational number $w\}$;
$S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $q ...

If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then :

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...

Let $W$ denote the words in the English dictionary. Define the relation $R$ by
$R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ 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 :

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 :