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 $\mathrm{m}+\mathrm{n}$ is equal to ___________.

Let a relation $R$ on $N \times N$ 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 $A=\{1,2,3,4,5,6,7\}$. Define $B=\{T \subseteq A$ : either $1 \notin T$ or $2 \in T\}$ and $C=\{T \subseteq A: T$ the sum of all the elements of $T$ is a prime number $\}$. Then the number of elements in the set $B \cup C$ is ________________.

Define a relation $R$ over a class of $n imes n$ real matrices $A$ and $B$ as "$ARB$ iff there exists a non-singular matrix $P$ such that $PAP^{-1} = B$". Then which of the following is true?

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 __________.