**Question:**

Determine whether each of the following relations are reflexive, symmetric and transitive:

(i)Relation R in the set *A* = {1, 2, 3…13, 14} defined as

$R=\{(x, y): 3 x-y=0\}$

(ii) Relation R in the set N of natural numbers defined as

$R=\{(x, y): y=x+5$ and $x<4\}$

(iii) Relation R in the set *A* = {1, 2, 3, 4, 5, 6} as

$R=\{(x, y): y$ is divisible by $x\}$

(iv) Relation R in the set Z of all integers defined as

$R=\{(x, y): x-y$ is as integer $\}$

(v) Relation R in the set *A* of human beings in a town at a particular time given by

(a) $R=\{(x, y): x$ and $y$ work at the same place $\}$

(b) $R=\{(x, y): x$ and $y$ live in the same locality $\}$

(c) $R=\{(x, y): x$ is exactly $7 \mathrm{~cm}$ taller than $y\}$

(d) $R=\{(x, y): x$ is wife of $y\}$

(e) $R=\{(x, y): x$ is father of $y\}$

**Solution:**

(i) *A* = {1, 2, 3 … 13, 14}

R = {(*x*, *y*): 3*x* − *y* = 0}

∴R = {(1, 3), (2, 6), (3, 9), (4, 12)}

$R$ is not reflexive since $(1,1),(2,2) \ldots(14,14) \notin R$.

Also, $R$ is not symmetric as $(1,3) \in R$, but $(3,1) \notin R .[3(3)-1 \neq 0]$

Also, $R$ is not transitive as $(1,3),(3,9) \in R$, but $(1,9) \notin R$.

$[3(1)-9 \neq 0]$

Hence, R is neither reflexive, nor symmetric, nor transitive.

(ii) $R=\{(x, y): y=x+5$ and $x<4\}=\{(1,6),(2,7),(3,8)\}$

It is seen that $(1,1) \notin R$.

∴R is not reflexive.

$(1,6) \in R$

But,

$(6,1) \notin \mathrm{R} .$

∴R is not symmetric.

Now, since there is no pair in $R$ such that $(x, y)$ and $(y, z) \in R$, then $(x, z)$ cannot belong to $R$.

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

(iii) *A* = {1, 2, 3, 4, 5, 6}

$R=\{(x, y): y$ is divisible by $x\}$

We know that any number (*x)* is divisible by itself.

$\Rightarrow(x, x) \in R$

∴R is reflexive.

Now,

$(2,4) \in R$ [as 4 is divisible by 2 ]

But,

$(4,2) \notin R .$ [as 2 is not divisible by 4$]$

∴R is not symmetric.

Let $(x, y),(y, z) \in R$. Then, $y$ is divisible by $x$ and $z$ is divisible by $y$.

∴*z* is divisible by *x*.

$\Rightarrow(x, z) \in R$

∴R is transitive.

Hence, R is reflexive and transitive but not symmetric.

(iv) $R=\{(x, y): x-y$ is an integer $\}$

Now, for every $x \in \mathbf{Z},(x, x) \in R$ as $x-x=0$ is an integer.

∴R is reflexive.

Now, for every $x, y \in Z$ if $(x, y) \in R$, then $x-y$ is an integer.

$\Rightarrow-(x-y)$ is also an integer.

$\Rightarrow(y-x)$ is an integer.

$\therefore(y, x) \in R$

∴R is symmetric.

Now,

Let $(x, y)$ and $(y, z) \in R$, where $x, y, z \in Z$.

$\Rightarrow(x-y)$ and $(y-z)$ are integers.

$\Rightarrow x-z=(x-y)+(y-z)$ is an integer.

$\therefore(x, z) \in R$

∴R is transitive.

Hence, R is reflexive, symmetric, and transitive.

(v) (a) R = {(*x*, *y*): *x* and *y* work at the same place}

$\Rightarrow(x, x) \in \mathrm{R}$

∴ R is reflexive.

If $(x, y) \in R$, then $x$ and $y$ work at the same place.

$\Rightarrow y$ and $x$ work at the same place.

$\Rightarrow(y, x) \in R$

∴R is symmetric.

Now, let $(x, y),(y, z) \in R$

$\Rightarrow x$ and $y$ work at the same place and $y$ and $z$ work at the same place.

$\Rightarrow x$ and $z$ work at the same place.

$\Rightarrow(x, z) \in R$

∴ R is transitive.

Hence, R is reflexive, symmetric, and transitive.

(b) R = {(*x*, *y*): *x* and *y* live in the same locality}

Clearly $(x, x) \in R$ as $x$ and $x$ is the same human being.

∴ R is reflexive.

If $(x, y) \in R$, then $x$ and $y$ live in the same locality.

$\Rightarrow y$ and $x$ live in the same locality.

$\Rightarrow(y, x) \in R$

∴R is symmetric.

Now, let $(x, y) \in R$ and $(y, z) \in R$.

$\Rightarrow x$ and $y$ live in the same locality and $y$ and $z$ live in the same locality.

$\Rightarrow x$ and $z$ live in the same locality.

$\Rightarrow(x, z) \in R$

∴ R is transitive.

Hence, R is reflexive, symmetric, and transitive.

(c) R = {(*x*, *y*): *x* is exactly 7 cm taller than *y*}

Now,

$(x, x) \notin R$

Since human being *x *cannot be taller than himself.

∴R is not reflexive.

Now, let $(x, y) \in R$.

$\Rightarrow x$ is exactly $7 \mathrm{~cm}$ taller than $y$.

Then, *y* is not taller than *x*.

$\therefore(y, x) \notin \mathrm{R}$

Indeed if *x* is exactly 7 cm taller than *y*, then *y* is exactly 7 cm shorter than *x*.

∴R is not symmetric.

Now,

Let $(x, y),(y, z) \in R$.

$\Rightarrow x$ is exactly $7 \mathrm{~cm}$ taller thany and $y$ is exactly $7 \mathrm{~cm}$ taller than $z$.

$\Rightarrow x$ is exactly $14 \mathrm{~cm}$ taller than $z$.

$\therefore(x, z) \notin \mathrm{R}$

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

(d) R = {(*x*, *y*): *x* is the wife of *y*}

Now,

$(x, x) \notin R$

Since *x* cannot be the wife of herself.

∴R is not reflexive.

Now, let $(x, y) \in R$

$\Rightarrow x$ is the wife of $y$

Clearly *y* is not the wife of *x*.

$\therefore(y, x) \notin \mathrm{R}$

Indeed if *x* is the wife of *y*, then *y* is the husband of *x*.

∴ R is not symmetric.

Let $(x, y),(y, z) \in R$

$\Rightarrow x$ is the wife of $y$ and $y$ is the wife of $z$.

This case is not possible. Also, this does not imply that *x* is the wife of *z*.

$\therefore(x, z) \notin \mathrm{R}$

∴R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

(e) $R=\{(x, y): x$ is the father of $y\}$

$(x, x) \notin \mathrm{R}$

As *x* cannot be the father of himself.

∴R is not reflexive.

Now, let $(x, y) \in R$.

$\Rightarrow x$ is the father of $y$.

$\Rightarrow y$ cannot be the father of $y$.

Indeed, *y* is the son or the daughter of *y.*

*$\therefore(y, x) \notin \mathbf{R}$*

*∴ R is not symmetric.*

Now, let $(x, y) \in R$ and $(y, z) \in R$.

$\Rightarrow x$ is the father of $y$ and $y$ is the father of $z$.

$\Rightarrow x$ is not the father of $z$.

Indeed *x* is the grandfather of *z*.

$\therefore(x, z) \notin \mathrm{R}$

∴R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

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