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

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 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 \( A = \{4, 5, 7\} \) 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:

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:

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 \( A = \{-2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4\} \). 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:

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

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

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

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 \( A=\{0,1,2,\ldots,9\} \). 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:

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 ________.

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 ________.