The relation $R$ defined on the set of natural numbers as $\{(a, b) : a\}$ differs from $b$ by $3\}$ is given by

In the set $A = \{1, 2, 3, 4, 5\}$,a relation $R$ is defined by $R = \{(x, y) | x, y \in A \text{ and } x < y\}$. Then $R$ is

Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$ be denoted by $S$,such that $R$ is reflexive and symmetric,and $R$ contains exactly $10$ elements. Then the number of elements in $S$ is $...$ .

The empty relation on a set $A$ is

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 $R_{1} = R_{2}$.

Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $R = \{(a, b) : a, b \in A, b \text{ is exactly divisible by } a\}$. Find the domain of $R$.