Let $\mathrm{R}=\left\{(x, y) \in \mathbf{N} \times \mathbf{N}: \log _{\mathrm{e}}(x+y) \leq 2\right\}$. Then the minimum number of elements, required to be added in $R$ to make it a transitive relation, is $\_\_\_\_$ .

Let $\mathrm{A}=\{1,4,7\}$ and $\mathrm{B}=\{2,3,8\}$. Then the number of elements, in the relation $R=\left\{\left(\left(a_1, b_1\right),\left(a_2, b_2\right)\right) \in((A \times B) \times(A \times B)): a_1+b_2\right.$ divides $\left.a_2+b_1\right\}$ is $\_\_\_\_$ .

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 _________.

Let S be the set of the first 11 natural numbers. Then the number of elements in $A=\{B \subseteq S: n(B) \geqslant 2$ and the product of all elements of $B$ is even $\}$ is $\_\_\_\_$ .