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The domain of \( f(x) = \sqrt{([x]) - 1} + \sqrt{(4 - [x])} \) is: (where \( [ \, ] \) represents the greatest integer function)

The number of elements in the set \( \{\, n \in \mathbb{Z} : |n^2 - 10n + 19| < 6 \,\} \) is __________.

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 ________.

If \( n(A) = 10 \) and \( n(B) = 12 \), then the number of values that \( n(A \Delta B) \) can take is: (where \( \Delta \) is symmetric difference).

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 \( A \) sum that \( R \subset S \) and the number of elements in \( S \) in \( n \). Then, the minimum value of \( n \) 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 relation, is ________ .

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:

Let \( A = \{1,6,11,16,\ldots\} \) and \( B = \{9,16,23,30,\ldots\} \) be the sets consisting of the first \( 2025 \) terms of two arithmetic progressions. Then \( n(A \cup B) \) is:

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:

Let \( A=\{1,2,3,4,5,6,7\} \). Then the relation \( R=\{(x,y)\in A\times A: x+y=7\} \) is:

Let \( S=\{1,\,2,\,3,\,4,\,5,\,6,\,9\} \). Then the number of elements in the set \( T=\{A\subseteq S:\ A\neq \phi \ \text{and the sum of all the elements of }A \text{ is not a multiple of }3\} \) is ______.

Define a relation \( R \) on the interval \( \left[0, \frac{\pi}{2}\right) \) by \( x \,R\, y \) if and only if \( \sec^2 x - \tan^2 y = 1 \). Then \( R \) is:

Let \( A=\{1,2,3,\ldots,20\} \). Let \( R_1 \) and \( R_2 \) two relation 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 ______.

Let the relation \( R \) on the set \( M = \{1, 2, 3, \ldots, 16\} \) be given by \( R = \{(x, y) : 4y = 5x - 3 , \; x , y \in M\} \). Then the minimum number of elements required to be added in \( R \), in order to make the relation symmetric, is equal to:

Let \( R \) be a relation defined on the set \( \{1,2,3,4\} \times \{1,2,3,4\} \) by \( R = \{((a,b),(c,d)) : 2a + 3b = 3c + 4d\} \). Then the number of elements in \( R \) is:

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:

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:

Among the relations:

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 ____________.

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 ________________.

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

Let \( S = \{1, 2, 3, \ldots, 10\} \). Suppose \( M \) is the set of all the subsets of \( S \), then the relation \( R = \{(A, B) : (A \cap B \ne \phi; \, A, B \in M)\} \) is:

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:
\( (S_1): \) The number of elements in \( R \) is \( 18 \), and
\( (S_2): \) The relation \( R \) is symmetric but neither reflexive nor transitive

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 ______________.

Let \( A = \{4, 5, 7\} \) and \( B = \{2, 4, 6\} \) be two sets and let \( R \) be a relation from \( A \) to \( B \) defined by \( R : \{(x, y) : x < y , \; x \in A , \; y \in B\} \), then the difference between the sum of elements of domain and range of \( R \) is:

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

The number of non-empty equivalence relations on the set \( \{1, 2, 3\} \) is:

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 \( \_\_\_\_\_\_\_\_\_\_ \).

Let \( A = \{1,2, 3, 4, 5\} \). Let \( R \) be a relation on \( A \) defined by \( xRy \) if and only if \( 4x \le 5y \). 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:

Let the number of elements in sets \( A \) and \( B \) be five and two respectively. Then the number of subsets of \( A \times B \) each having at least \( 3 \) and at most \( 6 \) elements is:

Which of the following is not correct for relation \( R \) on the set of real numbers?

Let \( A=\{1,2,3,\ldots,100\} \). Let \( R \) be a relation on \( A \) defined by \( (x,y)\in R \) if 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 __________.

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.

If relations \( R_1 \) and \( R_2 \) from set \( A \) to set \( B \) are defined as: \( R_1 = \{(1,2),(3,4),(5,6)\} \) and \( R_2 = \{(2,1),(4,3),(6,5)\} \), then \( n(A \times B) \) can be equal to:

A group of \( 40 \) students appeared in an examination of \( 3 \) subjects – Mathematics, Physics and Chemistry. It was found that all students passed in at least one of the subjects, \( 20 \) students passed in Mathematics, \( 25 \) students passed in Physics, \( 16 \) students passed in Chemistry, at most \( 11 \) students passed in both Mathematics and Physics, at most \( 15 \) students passed in both Physics and Chemistry, at most \( 15 \) students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is \( \_\_\_\_\_ \).

If \( A = \{x : x \text{ is a letter in word BELOW}\} \), \( B = \{x : x \text{ is a letter in word WOOL}\} \) and \( C = A - B \), then the number of subsets of \( C \) 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 ________.

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 ______.

The relation \( R = \{(x, y) : x, y \in \mathbb{Z} \ \text{and} \ x + y \ \text{is even}\} \) is:

Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \( (a_1,b_1) \, R \, (a_2,b_2) \Leftrightarrow b_1 = b_2 \).
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:

Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to:

If the mean and variance of six observations \( 7, 10, 11, 15, a, b \) are \( 10 \) and \( \dfrac{20}{3} \), respectively, then the value of \( |a-b| \) is equal to:

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 \( x_{1} \le x_{2} \) or \( y_{1} \le y_{2} \).
Consider the two statements:
(I) \( R \) is reflexive but not symmetric.
(II) \( R \) is transitive
Then which one of the following is true?

Let \( A = \{-2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4\} \). Let \( R \) be a relation on \( A \) defined by \( xRy \) if and only if \( 2x + y \leq 2 \). 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:

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

Let \( A = \{1, 2, 3, \ldots, 10\} \) and \( B = \left\{ \dfrac{m}{n} : m, n \in A,\, m < n \text{ and } \gcd(m,n) = 1 \right\}. \) Then \( n(B) \) is equal to:

An organization awarded \( 48 \) medals in event ‘\( A \)’, \( 25 \) in event ‘\( B \)’ and \( 18 \) in event ‘\( C \)’. If these medals went to total \( 60 \) men and only \( 5 \) men got medals in all the three events, then, how many received medals in exactly two of three events?

The number of relations, defined on the set \( \{a,b,c,d\} \), which are both reflexive and symmetric, is equal to:

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 \( (A\cap B^{c})\cup(B\cap A^{c})=\varnothing \) and \( A R_{2} B \) if \( A\cup B^{c}=B\cup A^{c} \), \( \forall A,B\in P(S) \). Then:

Let \( A = \{-3,-2,-1,0,1,2,3\} \). Let \( R \) be a relation on \( A \) defined by \( xRy \) if and only if \( 0 \le x^{2} + 2y \le 4 \). 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:

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:

What does the shaded portion of the Venn diagram given below represent?

Let \( A = \{(x,y) \in R \times R : |x + y| \ge 3\} \) and \( B = \{(x,y) \in R \times R : |x| + |y| \le 3\}. \)
If \( C = \{(x,y) \in A \cap B : x = 0 \text{ or } y = 0\} \) then \(\displaystyle \sum_{(x,y)\in C} |x + y| \) is:

If the difference between the number of subsets of the sets \( A \) and \( B \) is \( 120 \), then choose the INCORRECT option.

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:

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:

Let \( A=\{1,4,7\} \) and \( B=\{2,3,8\} \). Then the number of elements, in the relation \( R=\{((a_{1},b_{1}),(a_{2},b_{2}))\in(A\times B)\times(A\times B):a_{1}+b_{2}\text{ divides }a_{2}+b_{1}\} \) 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 element of \( T \) is a prime number\( \} \). Then the number of element in the set \( B \cup C \) is \( \_ \_ \_ \_ \_ \_ \)

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 _________.

Let \( A=\{0,1,2,\ldots,9\} \). Let \( R \) be a relation on \( A \) defined by \( (x,y)\in R \) if and only if \( |x-y| \) is a multiple of \( 3 \).
Given below are two statements:
Statement I: \( n(R)=36 \).
Statement II: \( R \) is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below:

Let \( A = \{2,3,5,7,9\} \). Let \( R \) be the relation on \( A \) defined by \( xRy \) if and only if \( 2x \le 3y \). 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 symmetric relation. Then \( l + m \) is equal to:

The probability that a relation \( R \) from \( \{x,y\} \) to \( \{x,y\} \) is both symmetric and transitive, is equal to:

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:

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:

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:

Let \( A=\{n\in \mathbb{N}\mid n^{2}\leq n+10{,}000\}, B=\{3k+1\mid k\in \mathbb{N}\} \) and \( C=\{2k\mid k\in \mathbb{N}\} \), then the sum of all the elements of the set \( A\cap (B-C) \) is equal to ____________.

Let \( A = \{2, 3, 4, 5, 6\} \). Let \( R \) be a relation on the set \( A \times A \) given by \( (x,y)R(z,w) \) if and only if \( x \) divides \( z \) and \( y \leq w \). Then the number of elements in \( R \) is ________.

For \( n \ge 2 \), 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 \( n\!\left(S_{5}\right) \) is equal to ____________.

Let a set \( A = A_1 \cup A_2 \cup \ldots \cup A_k \), where \( A_i \cap A_j = \phi \) for \( i \neq j \), \( 1 \le i, j \le k \). Define the relation \( R \) from \( A \) to \( A \) by \( R = \{(x,y): y \in A_i \text{, if and only if } x \in A_i, 1 \le i \le k \} \). Then, \( R \) is:

Let \( A = \{ x \in R : |x + 1| < 2 \} \) and \( B = \{ x \in R : |x - 1| \ge 2 \} \). Then which one of the following statements is NOT true?

Value of \( \left[ \frac{1}{2} \right] + \left[ \frac{1}{2} + \frac{1}{100} \right] + \left[ \frac{1}{2} + \frac{2}{100} \right] + \ldots + \left[ \frac{1}{2} + \frac{254}{100} \right] \) (where \( [\cdot] \) denotes greatest integer function) is ________.

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If \( A = \{x : x \text{ is a letter in word BELOW}\} \), \( B = \{x : x \text{ is a letter in word WOOL}\} \) and \( C = A - B \), then the number of subsets of \( C \) is: