Let $A = \{1, 2, 3\}$. What is the total number of relations defined on $A$?

Give an example of a relation which is reflexive and transitive but not symmetric.

Give an example of a relation on a set $A = \{4, 6, 8\}$ which is reflexive and symmetric but not transitive.

Let $R$ be a relation on $\mathbb{R}$,given by $R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number} \}$. Then $R$ is

Give an example of a relation that is symmetric and transitive but not reflexive.

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: