Question

Let $R$ be the relation on the set $R$ of all real numbers defined by $aRb$ if $|a-b|\le 1$. Then, $R$ is

• A. Reflexive and symmetric
• B. Symmetric only
• C. Transitive only
• D. Anti-symmetry only

Answer 1

The correct option is A Reflexive and symmetric

Given, $aRb$ if $|a-b|\le 1$

Now, $|a-a|=0<1$

$\therefore aRa,\forall a\in R$

Hence, $R$ is reflexive

Again, $aRb\Rightarrow |a-b|\le 1$

$\Rightarrow |b-a|\le 1$ ....... [Using property of modulus]

$\Rightarrow$ $bRa$

Hence, $R$ is symmetric.

We know that $1R\frac{1}{2}$ and $\frac{1}{2}R1$, but $\frac{1}{2}\ne 1$

Hence, $R$ is not anti-symmetric.

Further, $1R2$ i.e. $|1-2|=1$and $2R3$ i.e. $|2-3|=1$, but $1R3$

$[\because |1-3|=2>1]$

So, $R$ is not transitive.