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

Let a relation $R$ on $N \times N$ be defined as: $(x_1, y_1) R (x_2, y_2)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements:
$(I)$ $R$ is reflexive but not symmetric.
$(II)$ $R$ is transitive.
Then which one of the following is true?

Show that the number of equivalence relations on the set $A = \{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is $2$.

Let $R$ be the relation in the set $\{1, 2, 3\}$ given by $R = \{(1, 1), (2, 2), (3, 3)\}$. Choose the correct answer.

$A$ relation $\rho$ on the set of real numbers $\mathbb{R}$ is defined as $\{x \rho y : xy > 0\}$. Then,which of the following is/are true?

Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $R_{1}$ be a relation in $X$ given by $R_{1} = \{(x, y) : x - y \text{ is divisible by } 3\}$ and $R_{2}$ be another relation on $X$ given by $R_{2} = \{(x, y) : \{x, y\} \subset \{1, 4, 7\} \text{ or } \{x, y\} \subset \{2, 5, 8\} \text{ or } \{x, y\} \subset \{3, 6, 9\}\}$. Show that $R_{1} = R_{2}$.