Question

Let $R$ be a relation defined as $aRb$ if $\left|a\right|\le b.$ Then, the relation $R$ is

• A. reflexive
• B. symmetric
• C. transitive
• D. None of these

Answer 1

The correct option is C transitive

$R$ is not reflexive, if $-a$ is any negative real number, then $|-a|>-a$ so that $-a$ is not in $R-a.$

$R$ is not symmetric.

Consider the real numbers $a=-2$ and $b=3$.

Then, $aRb$ as $|-2|<3.$

But $b$ not in $Ra$ as $\left|3\right|\le -2.$

$R$ is transitive: let $a,b,c$ be three real numbers such that $\left|a\right|\le b$ and $\left|b\right|\le c.$

But $\left|a\right|\le b\Rightarrow b\ge 0,$ and so $\left|b\right|\le c\Rightarrow b\le c.$

So, it follows $\left|a\right|\le c.$