Sets and Relation(PYQ’s)

The number of relations on the set $A = \{1, 2, 3\}$ , containing at most 6 elements including $(1, 2)$ , which are reflexive and transitive but not symmetric, is.
For $n \geq 2$ , let $S_n$ denote the set of all subsets of $\{1, 2, \ldots, n\}$ with no two consecutive numbers. For example, $\{1, 3, 5\} \in S_6$ , but $\{1, 2, 4\} \notin S_6$ . Then $n(S_5)$ is equal to.
Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$ , where $P$ is the set of all possible products of distinct elements of $S$ . Then the number of all ordered pairs $(x, y)$ , $x \in S$ , $y \in A$ , such that $x$ divides $y$ , is.
Let $A = \{1, 2, 3\}$ . The number of relations on $A$ , containing $(1, 2)$ and $(2, 3)$ , which are reflexive and transitive but not symmetric, is.
Let $A = \{2, 3, 6, 7\}$ and $B = \{4, 5, 6, 8\}$ . Let $R$ be a relation defined on $A \times B$ by $(a_1, b_1) R (a_2, b_2)$ if and only if $a_1 + a_2 = b_1 + b_2$ . Then the number of elements in $R$ is \underline{\hspace{1cm}}.
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $m + n$ is equal to \underline{\hspace{1cm}}.
Let $A = \{1, 2, 3, \ldots, 20\}$ . Let $R_1$ and $R_2$ be two relations on $A$ such that
\[
$R_1 = \{(a, b) : b \text{ is divisible by } a\}$
\]
\[
$R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}$ .
\]
Then, number of elements in $R_1 - R_2$ is equal to \underline{\hspace{1cm}}.
Let $A = \{1, 2, 3, \ldots, 100\}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $2x = 3y$ . Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$ and the number of elements in $R_1$ is $n$ . Then, the minimum value of $n$ is \underline{\hspace{1cm}}.
Let $A = \{1, 2, 3, 4\}$ and $R = \{(1, 2), (2, 3), (1, 4)\}$ be a relation on $A$ . Let $S$ be the equivalence relation on $A$ such that $R \subset S$ and the number of elements in $S$ is $n$ . Then, the minimum value of $n$ is \underline{\hspace{1cm}}.
The number of symmetric relations defined on the set $\{1, 2, 3, 4\}$ which are not reflexive is \underline{\hspace{1cm}}.
The number of elements in the set
\[
$\{n \in \mathbb{N} : 10 \leq n \leq 100 \text{ and } 3^n - 3 \text{ is a multiple of } 7\}$
\]
is \underline{\hspace{1cm}}.
Let $A = \{1, 2, 3, 4\}$ and $R$ be a relation on the set $A \times A$ defined by
\[
$R = \{((a, b), (c, d)) : 2a + 3b = 4c + 5d\}$ .
\]
Then the number of elements in $R$ is \underline{\hspace{1cm}}.
Let $A = \{-4, -3, -2, 0, 1, 3, 4\}$ and $R = \{(a, b) \in A \times A : b = |a| \text{ or } b^2 = a + 1\}$ be a relation on $A$ . Then the minimum number of elements that must be added to the relation $R$ so that it becomes reflexive and symmetric is \underline{\hspace{1cm}}.
The number of relations on the set $\{1, 2, 3\}$ containing $(1, 2)$ and $(2, 3)$ , which are reflexive and transitive but not symmetric, is \underline{\hspace{1cm}}.
The number of elements in the set
\[
$\{n \in \mathbb{Z} : |n^2 - 10n + 19| < 6\}$
\]
is \underline{\hspace{1cm}}.
Let $A = \{0, 3, 4, 6, 7, 8, 9, 10\}$ and $R$ be the relation defined on $A$ such that $R = \{(x, y) \in A \times A : x - y \text{ is odd positive integer or } x - y = 2\}$ . The minimum number of elements that must be added to the relation $R$ , so that it is a symmetric relation, is equal to \underline{\hspace{1cm}}.
Let $A = \{1, 2, 3, 4, \ldots, 10\}$ and $B = \{0, 1, 2, 3, 4\}$ . The number of elements in the relation $R = \{(a, b) \in A \times A : 2(a - b)^2 + 3(a - b) \in B\}$ is \underline{\hspace{1cm}}.
Let $S = \{1, 2, 3, 5, 7, 10, 11\}$ . The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3 is \underline{\hspace{1cm}}.
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 relation is \underline{\hspace{1cm}}.
Let $S = \{4, 6, 9\}$ and $T = \{9, 10, 11, \ldots, 1000\}$ . If $A = \{a_1 + a_2 + \ldots + a_k : k \in \mathbb{N}, a_1, a_2, a_3, \ldots, a_k \in S\}$ , then the sum of all the elements in the set $T - A$ is equal to \underline{\hspace{1cm}}.
Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ and $B = \{3, 6, 7, 9\}$ . Then the number of elements in the set $\{C \subseteq A : C \cap B \neq \emptyset\}$ is \underline{\hspace{1cm}}.
Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ . Define $B = \{T \subseteq A : \text{either } 1 \notin T \text{ or } 2 \in T\}$ and $C = \{T \subseteq A : \text{the sum of all the elements of } T \text{ is a prime number}\}$ . Then the number of elements in the set $B \cup C$ is \underline{\hspace{1cm}}.
Let $R_1$ and $R_2$ be relations on the set $\{1, 2, \ldots, 50\}$ such that
\[
$R_1 = \{(p, p^n) : p \text{ is a prime and } n \geq 0 \text{ is an integer}\}$
\]
\[
$R_2 = \{(p, p^n) : p \text{ is a prime and } n = 0 \text{ or } 1\}$ .
\]
Then, the number of elements in $R_1 - R_2$ is \underline{\hspace{1cm}}.
Let $A = \{n \in \mathbb{N} : \text{H.C.F.}(n, 45) = 1\}$ and
\[
$B = \{2k : k \in \{1, 2, \ldots, 100\}\}$ .
\]
Then the sum of all the elements of $A \cap B$ is \underline{\hspace{1cm}}.
Let $A = \sum_{i=1}^{10} \sum_{j=1}^{10} \min \{i, j\}$ and $B = \sum_{i=1}^{10} \sum_{j=1}^{10} \max \{i, j\}$ . Then $A + B$ is equal to \underline{\hspace{1cm}}.
The sum of all the elements of the set $\{\alpha \in \{1, 2, \ldots, 100\} : \text{HCF}(\alpha, 24) = 1\}$ is \underline{\hspace{1cm}}.
If $A = \{x \in \mathbb{R} : |x - 2| > 1\}$ ,
\[
$B = \{x \in \mathbb{R} : \sqrt{x^2 - 3} > 1\}$ ,
\]
\[
$C = \{x \in \mathbb{R} : |x - 4| \geq 2\}$
\]
and $Z$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^c \cap Z$ is \underline{\hspace{1cm}}.
Let $A = \{n \in \mathbb{N} : n^2 \leq n + 10,000\}$ , $B = \{3k + 1 : k \in \mathbb{N}\}$ and $C = \{2k : k \in \mathbb{N}\}$ , then the sum of all the elements of the set $A \cap (B - C)$ is equal to \underline{\hspace{1cm}}.
Let $A = \{n \in \mathbb{N} : n \text{ is a 3-digit number}\}$
\[
$B = \{9k + 2 : k \in \mathbb{N}\}$
\]
and $C = \{9k + l : k \in \mathbb{N}\}$ for some $l (0 < l < 9)$
If the sum of all the elements of the set $A \cap (B \cup C)$ is $274 \times 400$ , then $l$ is equal to \underline{\hspace{1cm}}.
Set $A$ has $m$ elements and set $B$ has $n$ elements. If the total number of subsets of $A$ is 112 more than the total number of subsets of $B$ , then the value of $m \cdot n$ is \underline{\hspace{1cm}}.
Let $X = \{n \in \mathbb{N} : 1 \leq n \leq 50\}$ . If
\[
$A = \{n \in X : n \text{ is a multiple of } 2\}$
\]
and
\[
$B = \{n \in X : n \text{ is a multiple of } 7\}$ ,
\]
then the number of elements in the smallest subset of $X$ containing both $A$ and $B$ is \underline{\hspace{1cm}}.
Let $A = \{0, 1, 2, 3, 4, 5\}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $\max\{x, y\} \in \{3, 4\}$ . Then among the statements
\begin{enumerate}
\item[latex]\text{S}_1 [/latex]: The number of elements in $R$ is 18, and
\item[latex]\text{S}_2 [/latex]: The relation $R$ is symmetric but neither reflexive nor transitive
\end{enumerate}

(A) both are false
(B) only $(\text{S}_1)$ is true
(C) only $(\text{S}_2)$ is true
(D) both are true
Let $A = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : |\alpha - 1| \leq 4 \text{ and } |\beta - 5| \leq 6\}$ and $B = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \leq 144\}$ . Then
(A) $A \subset B$
(B) $B \subset A$
(C) neither $A \subset B$ nor $B \subset A$
(D) $A \cup B = \{(x, y) : -4 \leq x \leq 4, -1 \leq y \leq 11\}$
Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and $R$ be a relation on $A$ defined by $xRy$ if and only if $2x - y \in \{0, 1\}$ . Let $l$ be the number of elements in $R$ . Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations, respectively. Then $l + m + n$ is equal to:
(A) 17
(B) 18
(C) 15
(D) 16
Consider the sets $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}$ , $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}$ , $C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \leq 4\}$ , and $D = A \cap B$ . The total number of one-one functions from the set $D$ to the set $C$ is:
(A) 15120
(B) 18290
(C) 17160
(D) 19320
Let $A = \{-2, -1, 0, 1, 2, 3\}$ . Let $R$ be a relation on $A$ defined by $xRy$ if and only if $y = \max\{x, 1\}$ . Let $l$ be the number of elements in $R$ . Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations, respectively. Then $l + m + n$ is equal to:
(A) 11
(B) 12
(C) 14
(D) 13
Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$ . Let $R$ be a relation on $A$ defined by $xRy$ if and only if $0 \leq x^2 + 2y \leq 4$ . Let $l$ be the number of elements in $R$ and $m$ be the minimum number of elements required to be added in $R$ to make it a reflexive relation. Then $l + m$ is equal to:
(A) 18
(B) 20
(C) 17
(D) 19
Let $A = \{1, 2, 3, \ldots, 100\}$ and $R$ be a relation on $A$ such that $R = \{(a, b) : a = 2b + 1\}$ . Let $(a_1, a_2), (a_2, a_3), (a_3, a_4), \ldots, (a_k, a_{k+1})$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer $k$ , for which such a sequence exists, is equal to:
(A) 6
(B) 8
(C) 7
(D) 5
Let $A$ be the set of all functions $f : \mathbb{Z} \to \mathbb{Z}$ and $R$ be a relation on $A$ such that $R = \{(f, g) : f(0) = g(1) \text{ and } f(1) = g(0)\}$ . Then $R$ is:
(A) Symmetric and transitive but not reflexive
(B) Symmetric but neither reflexive nor transitive
(C) Transitive but neither reflexive nor symmetric
(D) Reflexive but neither symmetric nor transitive
Let $S = \mathbb{N} \cup \{0\}$ . Define a relation $R$ from $S$ to $\mathbb{R}$ by: $R = \{(x, y) : \log_e y = x \log_e \left(\frac{2}{5}\right), x \in S, y \in \mathbb{R}\}$ . Then, the sum of all the elements in the range of $R$ is equal to:
(A) $\frac{3}{2}$
(B) $\frac{10}{9}$
(C) $\frac{5}{2}$
(D) $\frac{5}{3}$
Define a relation $R$ on the interval $[0, \frac{\pi}{2})$ by $xRy$ if and only if $\sec^2 x - \tan^2 y = 1$ . Then $R$ is:
(A) both reflexive and symmetric but not transitive
(B) both reflexive and transitive but not symmetric
(C) reflexive but neither symmetric nor transitive
(D) an equivalence relation
The relation $R = \{(x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even}\}$ is:
(A) reflexive and transitive but not symmetric
(B) reflexive and symmetric but not transitive
(C) an equivalence relation
(D) symmetric and transitive but not reflexive
Let $A = \{ x \in (0, \pi) - \{\frac{\pi}{2}\} : \log_{\frac{2}{\pi}} |\sin x| + \log_{\frac{2}{\pi}} |\cos x| = 2 \}$ and $B = \{ x > 0 : \sqrt{x} (\sqrt{x} - 4) - 3\sqrt{x} - 2 + 6 = 0 \}$ . Then $n(A \cup B)$ is equal to:
(A) 4
(B) 8
(C) 6
(D) 2
Let $X = \mathbb{R} \times \mathbb{R}$ . Define a relation $R$ on $X$ as: $(a_1, b_1) R (a_2, b_2) \iff b_1 = b_2$ . \\
Statement I: $R$ is an equivalence relation. \\
Statement II: For some $(a, b) \in X$ , the set $S = \{(x, y) \in X : (x, y) R (a, b)\}$ represents a line parallel to $y = x$ . \\
In the light of the above statements, choose the correct answer from the options given below:

(A) Both Statement I and Statement II are true
(B) Statement I is true but Statement II is false
(C) Both Statement I and Statement II are false
(D) Statement I is false but Statement II is true
Let $X = \mathbb{R} \times \mathbb{R}$ . Define a relation $R$ on $X$ as: $(a_1, b_1) R (a_2, b_2) \iff b_1 = b_2$ . \\
Statement I: $R$ is an equivalence relation. \\
Statement II: For some $(a, b) \in X$ , the set $S = \{(x, y) \in X : (x, y) R (a, b)\}$ represents a line parallel to $y = x$ . \\
In the light of the above statements, choose the correct answer from the options given below:

(A) Both Statement I and Statement II are true
(B) Statement I is true but Statement II is false
(C) Both Statement I and Statement II are false
(D) Statement I is false but Statement II is true
Let $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\}$ and $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\}$ . If $C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\}$ , then $\sum_{(x,y) \in C} |x + y|$ is:
(A) 18
(B) 24
(C) 15
(D) 12
Let $R = \{(1, 2), (2, 3), (3, 3)\}$ be a relation defined on the set $\{1, 2, 3, 4\}$ . Then the minimum number of elements needed to be added in $R$ so that $R$ becomes an equivalence relation is:
(A) 9
(B) 8
(C) 7
(D) 10
Let $A = \{1, 2, 3, \ldots, 10\}$ and $B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}$ . Then $n(B)$ is equal to:
(A) 29
(B) 31
(C) 37
(D) 36
The number of non-empty equivalence relations on the set $\{1, 2, 3\}$ is:
(A) 7
(B) 4
(C) 5
(D) 6
Let $A = \{2, 3, 6, 8, 9, 11\}$ and $B = \{1, 4, 5, 10, 15\}$ . Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3ad - 7bc$ is an even integer. Then the relation $R$ is
(A) reflexive but not symmetric
(B) an equivalence relation
(C) reflexive and symmetric but not transitive
(D) transitive but not symmetric
Let $A = \{1, 2, 3, 4, 5\}$ . Let $R$ be a relation on $A$ defined by $x R y$ if and only if $4x \leq 5y$ . Let $m$ be the number of elements in $R$ and $n$ be the minimum number of elements from $A \times A$ that are required to be added to $R$ to make it a symmetric relation. Then $m + n$ is equal to:
(A) 23
(B) 26
(C) 25
(D) 24
Let $A = \{n \in [100, 700] \cap \mathbb{N} : n \text{ is neither a multiple of } 3 \text{ nor a multiple of } 4\}$ . Then the number of elements in $A$ is
(A) 300
(B) 310
(C) 290
(D) 280
Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, \ldots, 20\}$ be given by $R_1 = \{(x, y) : 2x - 3y = 2\}$ and
\[
$R_2 = \{(x, y) : -5x + 4y = 0\}$ .
\]
If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$ , respectively, in order to make the relations symmetric, then $M + N$ equals

(A) 16
(B) 12
(C) 8
(D) 10
Let a relation $R$ on $\mathbb{N} \times \mathbb{N}$ be defined as: $(x_1, y_1) R (x_2, y_2)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$ . Consider the two statements:
\item[(I)] $R$ is reflexive but not symmetric.
\item[(II)] $R$ is transitive

Then which one of the following is true?

(A) Only (II) is correct.
(B) Both (I) and (II) are correct.
(C) Neither (I) nor (II) is correct.
(D) Only (I) is correct.
Consider the relations $R_1$ and $R_2$ defined as $a R_1 b \Leftrightarrow a^2 + b^2 = 1$ for all $a, b \in \mathbb{R}$ and $(a, b) R_2 (c, d) \Leftrightarrow a + d = b + c$ for all $(a, b), (c, d) \in \mathbb{N} \times \mathbb{N}$ . Then:
(A) $R_1$ and $R_2$ both are equivalence relations
(B) Only $R_1$ is an equivalence relation
(C) Only $R_2$ is an equivalence relation
(D) Neither $R_1$ nor $R_2$ is an equivalence relation
Let $R$ be a relation on $\mathbb{Z} \times \mathbb{Z}$ defined by $(a, b) R (c, d)$ if and only if $ad - bc$ is divisible by 5. Then $R$ is
(A) Reflexive and transitive but not symmetric
(B) Reflexive and symmetric but not transitive
(C) Reflexive but neither symmetric nor transitive
(D) Reflexive, symmetric and transitive
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:
(A) 8
(B) 10
(C) 4
(D) 6
Let $S = \{1, 2, 3, \ldots, 10\}$ . Suppose $M$ is the set of all the subsets of $S$ , then the relation
\[
$R = \{(A, B) : A \cap B \neq \emptyset ; A, B \in M\}$
\]
is:

(A) symmetric only
(B) reflexive only
(C) symmetric and reflexive only
(D) symmetric and transitive only
Let $A = \{1, 3, 4, 6, 9\}$ and $B = \{2, 4, 5, 8, 10\}$ . Let $R$ be a relation defined on $A \times B$ such that $R = \{((a_1, b_1), (a_2, b_2)) : a_1 \leq b_2 \text{ and } b_1 \leq a_2\}$ . Then the number of elements in the set $R$ is:
(A) 180
(B) 26
(C) 52
(D) 160
An organization awarded 48 medals in event $\text{A}$ , 25 in event $\text{B}$ and 18 in event $\text{C}$ . If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?
(A) 10
(B) 15
(C) 21
(D) 9
Let $A = \{2, 3, 4\}$ and $B = \{8, 9, 12\}$ . Then the number of elements in the relation
\[
$R = \{((a_1, b_1), (a_2, b_2)) \in (A \times B, A \times B) : a_1 \text{ divides } b_2 \text{ and } a_2 \text{ divides } b_1\}$
\]
is:

(A) 18
(B) 24
(C) 36
(D) 12
Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ . Then the relation $R = \{(x, y) \in A \times A : x + y = 7\}$ is:
(A) reflexive but neither symmetric nor transitive
(B) transitive but neither symmetric nor reflexive
(C) symmetric but neither reflexive nor transitive
(D) an equivalence relation
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 $A R_1 B$ if $(A \cap B^c) \cup (B \cap A^c) = \emptyset$ and $A R_2 B$ if $A \cup B^c = B \cup A^c$ , $\forall A, B \in P(S)$ . Then:
(A) only $R_2$ is an equivalence relation
(B) both $R_1$ and $R_2$ are not equivalence relations
(C) both $R_1$ and $R_2$ are equivalence relations
(D) only $R_1$ is an equivalence relation
Let $R$ be a relation on $\mathbb{R}$ , given by $R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number}\}$ . Then $R$ is
(A) an equivalence relation
(B) reflexive and symmetric but not transitive
(C) reflexive and transitive but not symmetric
(D) reflexive but neither symmetric nor transitive
Among the relations
\[
$S = \{(a, b) : a, b \in \mathbb{R} - \{0\}, 2 + \frac{a}{b} > 0\}$
\]
and $T = \{(a, b) : a, b \in \mathbb{R}, a^2 - b^2 \in \mathbb{Z}\}$ ,

(A) $S$ is transitive but $T$ is not
(B) both $S$ and $T$ are symmetric
(C) neither $S$ nor $T$ is transitive
(D) $T$ is symmetric but $S$ is not
Let $R$ be a relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a, b) R (c, d)$ if and only if $ad(b - c) = bc(a - d)$ . Then $R$ is
(A) symmetric and transitive but not reflexive
(B) reflexive and symmetric but not transitive
(C) transitive but neither reflexive nor symmetric
(D) symmetric but neither reflexive nor transitive
The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c)\}$ on the set $\{a, b, c\}$ so that it becomes symmetric and transitive is:
(A) 7
(B) 3
(C) 4
(D) 5
Let $R$ be a relation defined on $\mathbb{N}$ as $a R b$ if $2a + 3b$ is a multiple of 5, $a, b \in \mathbb{N}$ . Then $R$ is
(A) an equivalence relation
(B) non reflexive
(C) symmetric but not transitive
(D) transitive but not symmetric
The relation $R = \{(a, b) : \gcd(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\}$ is:
(A) reflexive but not symmetric
(B) transitive but not reflexive
(C) symmetric but not transitive
(D) neither symmetric nor transitive
Let $R$ be a relation from the set $\{1, 2, 3, \ldots, 60\}$ to itself such that $R = \{(a, b) : b = pq, \text{ where } p, q \geq 3 \text{ are prime numbers}\}$ . Then, the number of elements in $R$ is:
(A) 600
(B) 660
(C) 540
(D) 720
For $\alpha \in \mathbb{N}$ , consider a relation $R$ on $\mathbb{N}$ given by $R = \{(x, y) : 3x + \alpha y \text{ is a multiple of } 7\}$ . The relation $R$ is an equivalence relation if and only if:
(A) $\alpha = 14$
(B) $\alpha$ is a multiple of 4
(C) 4 is the remainder when $\alpha$ is divided by 10
(D) 4 is the remainder when $\alpha$ is divided by 7
Let $R_1$ and $R_2$ be two relations defined on $\mathbb{R}$ by
\[
$a R_1 b \Leftrightarrow ab \geq 0$
\]
and $a R_2 b \Leftrightarrow a \geq b$
Then,

(A) $R_1$ is an equivalence relation but not $R_2$
(B) $R_2$ is an equivalence relation but not $R_1$
(C) both $R_1$ and $R_2$ are equivalence relations
(D) neither $R_1$ nor $R_2$ is an equivalence relation
Let a set $A = A_1 \cup A_2 \cup \ldots \cup A_k$ , where $A_i \cap A_j = \emptyset$ for $i \neq j, 1 \leq i, j \leq k$ . Define the relation $R$ from $A$ to $A$ by $R = \{(x, y) : y \in A_i \text{ if and only if } x \in A_i, 1 \leq i \leq k\}$ . Then, $R$ is:
(A) reflexive, symmetric but not transitive
(B) reflexive, transitive but not symmetric
(C) reflexive but not symmetric and transitive
(D) an equivalence relation
Let
\[
$R_1 = \{(a, b) \in \mathbb{N} \times \mathbb{N} : |a - b| \leq 13\}$
\]
and
\[
$R_2 = \{(a, b) \in \mathbb{N} \times \mathbb{N} : |a - b| \neq 13\}$ .
\]
Then on $\mathbb{N}$ :

(A) Both $R_1$ and $R_2$ are equivalence relations
(B) Neither $R_1$ nor $R_2$ is an equivalence relation
(C) $R_1$ is an equivalence relation but $R_2$ is not
(D) $R_2$ is an equivalence relation but $R_1$ is not
Which of the following is not correct for relation $R$ on the set of real numbers?
(A) $(x, y) \in R \Leftrightarrow 0 < |x| - |y| \leq 1$ is neither transitive nor symmetric
(B) $(x, y) \in R \Leftrightarrow 0 < |x - y| \leq 1$ is symmetric and transitive
(C) $(x, y) \in R \Leftrightarrow |x| - |y| \leq 1$ is reflexive but not symmetric
(D) $(x, y) \in R \Leftrightarrow |x - y| \leq 1$ is reflexive and symmetric
Out of all the patients in a hospital 89\% are found to be suffering from heart ailment and 98\% are suffering from lungs infection. If $K\%$ of them are suffering from both ailments, then $K$ can not belong to the set:
(A) $\{80, 83, 86, 89\}$
(B) $\{84, 86, 88, 90\}$
(C) $\{79, 81, 83, 85\}$
(D) $\{84, 87, 90, 93\}$
Let $\mathbb{N}$ be the set of natural numbers and a relation $R$ on $\mathbb{N}$ be defined by $R = \{(x, y) \in \mathbb{N} \times \mathbb{N} : x^3 - 3x^2 y - x y^2 + 3y^3 = 0\}$ . Then the relation $R$ is:
(A) symmetric but neither reflexive nor transitive
(B) reflexive but neither symmetric nor transitive
(C) reflexive and symmetric, but not transitive
(D) an equivalence relation
Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as
\[
$ARB \iff \text{there exists a non-singular matrix } P \text{ such that } PAP^{-1} = B$ .
\]
Then which of the following is true?

(A) $R$ is reflexive, transitive but not symmetric
(B) $R$ is symmetric, transitive but not reflexive
(C) $R$ is reflexive, symmetric but not transitive
(D) $R$ is an equivalence relation
In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?
(A) $Q$ and $R$
(B) None of these
(C) $P$ and $R$
(D) $P$ and $Q$
Let $A = \{2, 3, 4, 5, \ldots, 30\}$ 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:
(A) 5
(B) 6
(C) 8
(D) 7
The number of elements in the set $\{x \in \mathbb{R} : (|x| - 3)|x + 4| = 6\}$ is equal to:
(A) 4
(B) 2
(C) 3
(D) 1
Let $R = \{(P, Q) \mid P \text{ and } Q \text{ are at the same distance from the origin}\}$ be a relation, then the equivalence class of $(1, -1)$ is the set:
(A) $S = \{(x, y) \mid x^2 + y^2 = \sqrt{2}\}$
(B) $S = \{(x, y) \mid x^2 + y^2 = 2\}$
(C) $S = \{(x, y) \mid x^2 + y^2 = 1\}$
(D) $S = \{(x, y) \mid x^2 + y^2 = 4\}$
A survey shows that 73\% of the persons working in an office like coffee, whereas 65\% like tea. If $x$ denotes the percentage of them who like both coffee and tea, then $x$ cannot be:
(A) 63
(B) 36
(C) 54
(D) 38
Let $\bigcup_{i=1}^{50} X_i = \bigcup_{i=1}^n Y_i = T$ 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:
(A) 30
(B) 50
(C) 15
(D) 45
A survey shows that 63\% of the people in a city read newspaper $A$ whereas 76\% read newspaper $B$ . If $x\%$ of the people read both the newspapers, then a possible value of $x$ can be:
(A) 37
(B) 65
(C) 29
(D) 55
Let $R_1$ and $R_2$ be two relations defined as follows:
\[
$R_1 = \{(a, b) \in \mathbb{R}^2 : a^2 + b^2 \in \mathbb{Q}\}$
\]
and
\[
$R_2 = \{(a, b) \in \mathbb{R}^2 : a^2 + b^2 \notin \mathbb{Q}\}$ ,
\]
where $\mathbb{Q}$ is the set of all rational numbers. Then:

(A) Neither $R_1$ nor $R_2$ is transitive
(B) $R_2$ is transitive but $R_1$ is not transitive
(C) $R_1$ and $R_2$ are both transitive
(D) $R_1$ is transitive but $R_2$ is not transitive
Consider the two sets:
\[
$A = \{m \in \mathbb{R} : \text{both the roots of } x^2 - (m + 1)x + m + 4 = 0 \text{ are real}\}$
\]
and $B = [-3, 5)$ .
Which of the following is not true?

(A) $A \cap B = \{-3\}$
(B) $B - A = (-3, 5)$
(C) $A \cup B = \mathbb{R}$
(D) $A - B = (-\infty, -3) \cup (5, \infty)$
If $R = \{(x, y) : x, y \in \mathbb{Z}, x^2 + 3y^2 \leq 8\}$ is a relation on the set of integers $\mathbb{Z}$ , then the domain of $R^{-1}$ is:
(A) $\{0, 1\}$
(B) $\{-2, -1, 1, 2\}$
(C) $\{-1, 0, 1\}$
(D) $\{-2, -1, 0, 1, 2\}$
If $A = \{x \in \mathbb{R} : |x| < 2\}$ and $B = \{x \in \mathbb{R} : |x - 2| \geq 3\}$ ; then:
(A) $A - B = [-1, 2)$
(B) $A \cup B = \mathbb{R} - (2, 5)$
(C) $A \cap B = (-2, -1)$
(D) $B - A = \mathbb{R} - (-2, 5)$
Let $A, B$ and $C$ be sets such that $\emptyset \neq A \cap B \subseteq C$ . Then which of the following statements is not true?
(A) If $(A - B) \subseteq C$ , then $A \subseteq C$
(B) $B \cap C \neq \emptyset$
(C) $(C \cup A) \cap (C \cup B) = C$
(D) If $(A - C) \subseteq B$ , then $A \subseteq B$
Two newspapers $A$ and $B$ are published in a city. It is known that 25\% of the city population 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 $B$ but not $A$ also look into advertisements, while 50\% of those who read both $A$ and $B$ look into advertisements. Then the percentage of the population who look into advertisements is:
(A) 13.5
(B) 13
(C) 12.8
(D) 13.9
Let $\mathbb{Z}$ be the set of integers.
\[
$A = \{x \in \mathbb{Z} : 2^{x + 2}(x^2 - 5x + 6) = 1\}$
\]
and
\[
$B = \{x \in \mathbb{Z} : -3 < 2x - 1 < 9\}$ ,
\]
then the number of subsets of the set $A \times B$ is

(A) $2^{12}$
(B) $2^{18}$
(C) $2^{10}$
(D) $2^{15}$
Let $S = \{1, 2, 3, \ldots, 100\}$ . The number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is:
(A) $2^{50} - 1$
(B) $2^{50}(2^{50} - 1)$
(C) $2^{100} - 1$
(D) $2^{50} + 1$
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 students who did not opt for any of the three courses is
(A) 42
(B) 102
(C) 1
(D) 38
Let $\mathbb{N}$ denote the set of all natural numbers. Define two binary relations on $\mathbb{N}$ as $R_1 = \{(x, y) \in \mathbb{N} \times \mathbb{N} : 2x + y = 10\}$ and $R_2 = \{(x, y) \in \mathbb{N} \times \mathbb{N} : x + 2y = 10\}$ . Then:
(A) Range of $R_1$ is $\{2, 4, 8\}$
(B) Range of $R_2$ is $\{1, 2, 3, 4\}$
(C) Both $R_1$ and $R_2$ are symmetric relations
(D) Both $R_1$ and $R_2$ are transitive relations
Two sets $A$ and $B$ are as under:
\[
$A = \{(a, b) \in \mathbb{R} \times \mathbb{R} : |a - 5| < 1 \text{ and } |b - 5| < 1\}$ ;
\]
\[
$B = \{(a, b) \in \mathbb{R} \times \mathbb{R} : 4(a - 6)^2 + 9(b - 5)^2 \leq 36\}$ ;
\]
Then

(A) neither $A \subset B$ nor $B \subset A$
(B) $B \subset A$
(C) $A \subset B$
(D) $A \cap B = \emptyset$ (an empty set)
Consider the following two binary relations on the set $A = \{a, b, c\}$ :
\[
$R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}$
\]
and
\[
$R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}$ .
\]
Then:

(A) both $R_1$ and $R_2$ are not symmetric
(B) $R_1$ is not symmetric but it is transitive
(C) $R_2$ is symmetric but it is not transitive
(D) both $R_1$ and $R_2$ are transitive
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

(A) $P \subset Q$ and $Q - P \neq \emptyset$
(B) $Q \not\subset P$
(C) $P \not\subset Q$
(D) $P = Q$
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 at least three elements are
(A) 219
(B) 256
(C) 275
(D) 510
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:
(A) $3^5$
(B) $2^5$
(C) $5^3$
(D) $5^2$
Let $R$ be the set of real numbers.
\[
\text{Statement I: } $A = \{(x, y) \in R \times R : y - x \text{ is an integer}\}$ \text{ is an equivalence relation on } $R$ .
\]
\[
\text{Statement II: } $B = \{(x, y) \in R \times R : x = \alpha y \text{ for some rational number } \alpha\}$ \text{ is an equivalence relation on } $R$ .
\]

(A) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I
(B) Statement I is true, Statement II is false
(C) Statement I is false, Statement II is true
(D) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I
Consider the following relations
\[
$R = \{(x, y) \mid x, y \text{ are real numbers and } x = wy \text{ for some rational number } w\}$
\]
\[
$S = \left\{ \left( \frac{m}{n}, \frac{p}{q} \right) \mid m, n, p \text{ and } q \text{ are integers such that } n, q \neq 0 \text{ and } qm = pn \right\}$ .
\]
Then

(A) $R$ is an equivalence relation but $S$ is not an equivalence relation
(B) Neither $R$ nor $S$ is an equivalence relation
(C) $S$ is an equivalence relation but $R$ is not an equivalence relation
(D) $R$ and $S$ both are equivalence relations
If $A, B$ and $C$ are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then:
(A) $A = C$
(B) $B = C$
(C) $A \cap B = \emptyset$
(D) $A = B$
Let $\mathbb{R}$ be the real line. Consider the following subsets of the plane $\mathbb{R} \times \mathbb{R}$ :
\[
$S = \{(x, y) : y = x + 1 \text{ and } 0 < x < 2\}$
\]
\[
$T = \{(x, y) : x - y \text{ is an integer}\}$ ,
\]
Which one of the following is true?

(A) Neither $S$ nor $T$ is an equivalence relation on $\mathbb{R}$
(B) Both $S$ and $T$ are equivalence relations on $\mathbb{R}$
(C) $S$ is an equivalence relation on $\mathbb{R}$ but $T$ is not
(D) $T$ is an equivalence relation on $\mathbb{R}$ but $S$ is not
Let $W$ denote the words in the English dictionary. Define the relation $R$ by
\[
$R = \{(x, y) \in W \times W \mid \text{the words } x \text{ and } y \text{ have at least one letter in common}\}$ .
\]
Then, $R$ is

(A) reflexive, symmetric and not transitive
(B) reflexive, symmetric and transitive
(C) reflexive, not symmetric and transitive
(D) not reflexive, symmetric and transitive
Let be a relation on the set . The relation is:
(A) reflexive and symmetric only
(B) an equivalence relation
(C) reflexive only
(D) reflexive and transitive only