If $R = \{(6, 6), (9, 9), (6, 12), (12, 12), (12, 6)\}$ is a relation on set $A = \{3, 6, 9, 12\}$,then relation $R$ is

Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(x, y) \mid x, y \in R \text{ and } xy \text{ is an irrational number}\}$. Then,the relation $r$ is:

The number of reflexive relations on a set with $4$ elements is equal to

Given a non-empty set $X$,consider $P(X)$ which is the set of all subsets of $X$. Define the relation $R$ in $P(X)$ as follows: For subsets $A, B$ in $P(X)$,$ARB$ if and only if $A \subset B$. Is $R$ an equivalence relation on $P(X)$? Justify your answer.

Let $R$ be a relation from $N$ to $N$ defined by $R = \{(a, b) : a, b \in N \text{ and } a = b^2\}$. Is the following statement true?
$(a, a) \in R$,for all $a \in N$

Let $A = \{1, 2, 3\}$. The relation $R$ on set $A$ is defined as $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$. Determine the nature of the relation $R$.