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 $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $xv = yu$. Show that $R$ is an equivalence relation.

Let $A = \{1, 2, 3\}$. The number of relations on $A$ containing $(1, 2)$ which are symmetric and transitive but not reflexive is . . . . . . .

Consider the relation $R$ on the set $\{-2, -1, 0, 1, 2\}$ defined by $(a, b) \in R$ if and only if $1 + ab > 0$. Then, among the statements:
$I$. The number of elements in $R$ is $17$
$II$. $R$ is an equivalence relation

Let $N$ be the set of natural numbers greater than $100$. Define the relation $R$ on $N$ by: $R = \{(x, y) \in N \times N : \text{the numbers } x \text{ and } y \text{ have at least two common divisors}\}.$ Then $R$ is-

If $R_{1}$ and $R_{2}$ are equivalence relations in a set $A$,show that $R_{1} \cap R_{2}$ is also an equivalence relation.