# Let A be the set of all human beings in a town at a particular time.

Question:

Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive:

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

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

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

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

Solution:

(i) Reflexivity:

Let $x$ be an arbitrary element of $R$. Then,

$x \in R$

$\Rightarrow x$ and $x$ work at the same place is true since they are the same.

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

So, $R$ is a reflexive relation.

Symmetry:

Let $(x, y) \in R$

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

$\Rightarrow y$ and $x$ work at the same place $\Rightarrow(y, x) \in R$

So, $R$ is a symmetric relation.

Transitivity:

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

$x$ and $y$ work at the same place. $y$ and $z$ also work at the same place.

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

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

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

So, $R$ is a transitive relation.

(ii) Reflexivity:

Let $x$ be an arbitrary element of $R$. Then,

$x \in R$

$\Rightarrow x$ and $x$ live in the same locality is true since they are the same.

So, $R$ is a reflexive relation.

Symmetry

Let $(x, y) \in R$

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

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

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

So, $R$ is a symmetric relation.

Transitivity:

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

$x$ and $y$ live in the same locality and $y$ and $z$ live in the same locality

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

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

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

So, $R$ is a transitive relation.

(iii)

Reflexivity:

Let $x$ be an element of $R$. Then, $x$ is wife of $x$ cannot be true. $\Rightarrow(\mathrm{x}, \mathrm{x}) \notin \mathrm{R}$

So, $R$ is not a reflexive relation.(iii)

Reflexivity:

Let $x$ be an element of $R$.

Then, $x$ is wife of $x$ cannot be true.

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

So, $R$ is not a reflexive relation.

Symmetry:

Let $(x, y) \in R$

$\Rightarrow x$ is wife of $y$

$\Rightarrow x$ is female and $y$ is male

$\Rightarrow y$ cannot be wife of $x$ as $y$ is husband of $x$

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

So, $R$ is not a symmetric relation.

Transitivity:

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

Since $x$ is wife of $y$, but $y$ cannot be the wife of $z, y$ is husband of $x$.

$\Rightarrow x$ is not the wife of $z$

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

So, $R$ is a transitive relation.

(iv)
Reflexivity:

Let $x$ be an arbitrary element of $R$. Then,

$x$ is father of $x$ cannot be true since no one can be father of himself.

So, $R$ is not a reflexive relation.

Symmetry:

Let $(x, y) \in R$

$\Rightarrow x$ is father of $y$

$\Rightarrow y$ is son/daughter of $x$

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

So, $R$ is not a symmetric relation.

Transitivity:

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

$x$ is father of $y$ and $y$ is father of $z$

$\Rightarrow x$ is grandfather of $z$

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

So, $R$ is not a transitive relation.