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.n is ______.

Let $X = R \times R$. 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 $A = {x \in (0, \pi) - {\frac{\pi}{2}} : \log_{(2/\pi)} |\sin x| + \log_{(2/\pi)} |\cos x| = 2}$ and $B = {x \ge 0 : x(x-4) - 3|x-2| + 6 = 0}$. Then $n(A \cup B)$ is equal to :

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 $A = { (\alpha, \beta ) \in R \times R : |\alpha - 1| \leq 4$ and $|\beta - 5| \leq 6 }$ and $B = { (\alpha, \beta ) \in R \times R : 16(\alpha - 2)^{2}+ 9(\beta - 6)^{2} \leq 144 }$. Then