Question

Let $R$ be a relation defined as $aRb$ if $1+ab>0$, then the relation $R$ is

• A. reflexive and symmetric
• B. symmetric but not reflexive
• C. transitive
• D. equivalence

Answer 1

The correct option is A reflexive and symmetric

Given relation is $aRb$ is $1+ab>0$,

Considering both $a$ and $b$ are real numbers,

We know that $ab=ba$,

$⟹aRb=1+ab>0=1+ba>0=bRa$,

$\therefore$ $R$ is a symmetric relation.

Now, $aRa=1+a^2$ as $a^2$ is always a positive real number

$\therefore 1+a^2>0$

$\therefore$ $R$ is a reflexive relation.

Now consider $aRb$ which implies $1+ab>0$ and also $bRc$ which implies $1+bc>0$

If we take $a=0.5,$ $b=-0.5$ and $c=-4$, then

$1+\left(0.5\right)(-0.5)=0.75>0$ and $1+(-0.5)(-4)=3>$

$\Rightarrow$ Both $aRb$ and $bRc$ are satisfied

But, $aRc=1+\left(0.5\right)(-4)=-2<0$

$\therefore$ $aRc$ is not a relation

Hence $R$ is not a equivalence relation, but is a reflexive and symmetric relation.