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

Let $R = \{(1, 3), (2, 2), (3, 2)\}$ and $S = \{(2, 1), (3, 2), (2, 3)\}$ be two relations on the set $A = \{1, 2, 3\}$. Find $R \circ S^{-1}$.

Let $R$ be a relation over the set $N \times N$ and it is defined by $(a, b)R(c, d) \iff a + d = b + c$. Then $R$ is

Let $S$ be the set of all real numbers. Then the relation $R = \{(a, b) : 1 + ab > 0\}$ on $S$ is

Let $R$ be a relation from $Q$ to $Q$ defined by $R = \{(a, b) : a, b \in Q \text{ and } a - b \in Z\}$. Show that $(a, b) \in R$ implies that $(b, a) \in R$.

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $Z$ of all integers defined as $R = \{(x, y) : x - y \text{ is an integer}\}$