Answer to Question #222564 in Discrete Mathematics for mostafa

Question

let z be the set of integers and R be the relation on Z defined as: aRb if and only if 1+ab>0 then

Answer 1

Given relation is $aRb$ is $1+ab>0.$

Considering both $a$ and $b$ are real numbers, we know that $ab=ba$

$$aRb=1+ab>0=>bRa=1+ba=1+ab>0$$

Then $R$ is a symmetric relation.

$$aRa=1+a^2>0$$

Then $R$ is a reflexive relation.

Let $a=0.5, b=-0.5,$ and $c=-4.$ Then

$$aRb=1+ab=1+0.5(-0.5)=0.75>0$$

$$bRc=1+bc=1+(-0.5)(-4)=3>0$$

But

$$aRc=1+ac=1+0.5(-4)=-1<0$$

$aRc$ is not a relation.

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

reflexive and symmetric relation.