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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:

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 \( A=\{1,2,3,\ldots,20\} \). 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 \( R \) 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:

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:

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 \( A=\{1,2,3,\ldots,100\} \). 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.

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

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 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 \( A \) 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: