Let $S$ be the set of all real numbers. $A$ relation $R$ has been defined on $S$ by $a R b \Leftrightarrow |a-b| \leq 1$. Then $R$ is:

On the set $N$ of natural numbers,the relation $R$ is defined by $nRm$ if $n$ is a factor of $m$ (i.e.,$n|m$). 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 $...$ .

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time given by
$R = \{(x, y) : x \text{ is the wife of } y\}$

Let $R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$ be a relation on the set $A = \{1, 2, 3, 4\}$. The relation $R$ 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 $R_{1} = R_{2}$.