Question

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

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

Answer 1

Answer: (C)

$(a,a)\in RR$ is reflexive
$(a,b)\in R\Rightarrow a$ lives within one kilometer from $b$
$\Rightarrow b$ lives within one kilometer from a
$\Rightarrow bRa$
$\therefore (a,b)\in R\Rightarrow (b,a)\in R$
$\therefore R$ is symmetric
$\therefore R$ is anti symmetric
because $(a,b)\in R\&(b,a)\in R$
and $a\ne b$