(i) Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2, \ldots \ldots, 14\}$ defined as $\mathrm{R}$ $=\{(\mathrm{x}, \mathrm{y}): 3 \mathrm{x}-\mathrm{y}=0\}$
(a) Put $\mathrm{y}=\mathrm{x}, 3 \mathrm{x}-\mathrm{x} \neq 0 \Rightarrow \mathrm{R}$ is not reflexive.
(b) If $3 x-y=0$, then $3 y-x \neq 0$, $R$ is not symmetric
(c) If $3 x-y=0,3 y-z=0$, then $3 x-z \neq 0$, $R$ is not transitive.
(ii) Relations in the set $\mathrm{N}$ of natural numbers in defined by $R=\{(x, y): y=x+5$ and $x < 4\}$
(a) Putting $y=x, x \neq x+5, R$ is not reflexive
(b) Putting $y=x+5$, then $x \neq y+5$, $R$ is not symmetric.
(c) If $y=x+5, z=y+5$, then $z \neq x+5 \Rightarrow R$ is not transitive.
(iii) Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3,4,5,6\}$ as $\mathrm{R}=\{(\mathrm{x}, \mathrm{y})$ : $\mathrm{y}$ is divisible by $\mathrm{x}\}$
(a) Putting $\mathrm{y}=\mathrm{x}, \mathrm{x}$ is divisible by $\mathrm{x} \Rightarrow \mathrm{R}$ is reflexive.
(b) If $y$ is divisible by $x$, then $x$ is not divisible by $y$ when $x \neq y \Rightarrow R$ is not symmetric.
(c) If $y$ is divisible by $x$ and $z$ is divisible by $y$ then $z$ is divisible by $x$ e.g., 2 is divisible by 1,4 is divisible by 2 .
$\Rightarrow 4$ is divisible by $1 \Rightarrow R$ is transitive.
(iv) Relation $\mathrm{R}$ in $\mathrm{Z}$ of all integers defined as $\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{x}-\mathrm{y}$ is an integer $\}$
(a) $x-x=0$ is an integer $\Rightarrow R$ is reflexive
(b) $x-y$ is an integer so is $y-x \Rightarrow R$ is transitive.
(c) $x-y$ is an integer, $y-z$ is an integer and $x-z$ is also an integer $\Rightarrow R$ is transitive.
(v) $\mathrm{R}$ is a set of human beings in a town at a particular time.
(a) $\mathrm{R}=\{(\mathrm{x}, \mathrm{y})\}: \mathrm{x}$ and $y$ work at the same place. It is reflexive as $x$ works at the same place. It is symmetric since $\mathrm{x}$ and $\mathrm{y}$ or $\mathrm{y}$ and $\mathrm{x}$ work at same place.
It is transitive since $x, y$ work at the same place and if $y, z$ work at the same place, then $x$ and $z$ also work at the same place.
(b) $R:\{(x, y): x$ and $y$ line in the same locality $\}$ With similar reasoning as in part (a), $\mathrm{R}$ is reflexive, symmetrical and transitive.
(c) $\mathrm{R}$ : $\{(\mathrm{x}, \mathrm{y})\}$ : $\mathrm{x}$ is exactly $7 \mathrm{~cm}$ taller than $y$ it is not reflexive: $x$ cannot $7 \mathrm{~cm}$ taller than $x$. It is not symmetric : $x$ is exactly $7 \mathrm{~cm}$ taller than $y, y$ cannot be exactly $7 \mathrm{~cm}$ taller than $x$. It is not transitive: If $x$ is exactly $7 \mathrm{~cm}$ taller than $y$ and if $y$ is exactly taller than $\mathrm{z}$, then $\mathrm{x}$ is not exactly $7 \mathrm{~cm}$ taller than $z$.
(d) $R=\{(x, y)$ : $x$ is wife of $y\}$
$R$ is not reflexive : $x$ cannot be wife of $x$. $R$ is not symmetric: $x$ is wife of $y$ but $y$ is not wife of $x$.
$R$ is not transitive : if $x$ is a wife of $y$ then $y$ cannot be the wife of anybody else.
(e) $R=\{(x, y): x$ is a father of $y\}$
It is not reflexive : $x$ cannot be father of himself. It is not symmetric : $x$ is a father of $y$ but $y$ cannot be the father of $x$.
It is not transitive : $\mathrm{x}$ is a father of $\mathrm{y}$ and $\mathrm{y}$ is a father of $z$ then $x$ cannot be the father of $z$.