Question

Relation R in the set A of human beings in a town at a particular time given by $R=\left\{\right(x,y):xisexactly7cmtallerthany\}$

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-Neither reflexive, nor symmetric, nor transitive

Answer 1

$R=\left\{\right(x,y):\text{x is exactly 7 cm taller than y}\}$

Now,

$(x,x)\notin R$

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

$\therefore R$ is not reflexive.

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

$\Rightarrow x$ is exactly $7$cm taller than $y$

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

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

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

$\therefore R$ is not symmetric.

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

$\Rightarrow$ $x$ is exactly $7$cm taller than $y$ and $y$ is exactly $7$cm taller than$z$.

$\Rightarrow$ $x$ is exactly $14$ taller than $z$.

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

$\therefore R$ is not transitive.

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