The relation "less than" in the set of natural numbers is

The number of relations on the set $A = \{1, 2, 3\}$ containing at most $6$ elements including $(1, 2)$,which are reflexive and transitive but not symmetric,is . . . . . . .

The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c), (b, d)\}$ on the set $\{a, b, c, d\}$ so that it is an equivalence relation,is $.........$

Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $R = \{(a, b) : a, b \in A, b \text{ is exactly divisible by } a\}$. Find the domain of $R$.

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 $S$ be the set of all real numbers. Then the relation $R = \{(a, b) : 1 + ab > 0\}$ on $S$ is