Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(a,b) \mid a,b \in R \text{ and } a - b + \sqrt{3} \text{ is an irrational number} \}$. The relation $r$ is

Let $R$ be an equivalence relation defined on a set containing $6$ elements. The minimum number of ordered pairs that $R$ should contain is

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

Show that the relation $R$ in the set $\{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1)\}$ is symmetric but neither reflexive nor transitive.

Let $A = \{1, 2, 3, 4\}$ and let $R = \{(2, 2), (3, 3), (4, 4), (1, 2)\}$ be a relation on $A$. Then $R$ is

Let $A = \{0, 1, 2, \ldots, 9\}$. Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $|x - y|$ is a multiple of $3$. Given below are two statements:
Statement $I$: $n(R) = 36$
Statement $II$: $R$ is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below: