Question

Which of the following statements is not correct for the relation $R$ defined by $aRb$, if and only if $b$ lives within one kilometre from $a$?

• A. $R$ is reflexive
• B. $R$ is symmetric
• C. $R$ is not anti-symmetric
• D. None of the above

Answer 1

The correct option is C

$R$ is not anti-symmetric

Explanation for correct option:

Anti symmetric: If $\left(x,y\right)\in R$ where $x\ne y$, then $\left(y,x\right)\notin R$

From the symmetric condition

$a$ lives within one kilometer from $b$

i.e., $(a,b)\in R$

$\Rightarrow b$ lives within one kilometer from $a$

i.e., $(b,a)\in R$

Since $(a,b)\in R$ and $(b,a)\in R$ holds when $a\ne b$.

So, the anti-symmetric does not hold.

Therefore, the final answer is option (c).

Explanation for incorrect options:

The given condition $aRb$ happens if and only if $b$ lives within one kilometer from $a$

For option (a)

Reflexive: A relation $R$ defined on a set $A$is called reflexive, if $a$ is an element of set $A$ then $\left(a,a\right)\in R$.
$a$ lives within one kilometer from $a$. so,
$(a,a)\in R$

i.e., $aRa$

Therefore $R$ is reflexive.

For option (b)

Symmetric: If $\left(x,y\right)\in R$ where $x\ne y$, then $\left(y,x\right)\in R$

If, $a$ lives within one kilometer from $b$

i.e., .$(a,b)\in R$

$\Rightarrow$ $b$ lives within one kilometer from $a$

$\therefore (b,a)\in R$

Therefore $R$ is symmetric.

Hence option (c) is the correct option.