Question

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

• A. Reflexive and symmetric
• B. Symmetric only
• C. Transitive only
• D. Antisymmetric only

Answer 1

The correct option is A

Reflexive and symmetric

Explanation For The Correct Option:

Determining the correct option

The given relation

$aRb$ iff $|a–b|\le 1$

Checking Reflexivity

$aRa\Rightarrow |a–a|=0<1$

So, $R$ is reflexive.

Checking Symmetricity

$aRb\Rightarrow |a–b|\le 1\Rightarrow |b–a|\le 1\Rightarrow bRa$

So, $R$ is symmetric.

Checking transitivity

If $1R2\Rightarrow \left|1-2\right|=1\le 1$

And $2R3\Rightarrow \left|3-2\right|=1\le 1$

Now for $1R3\Rightarrow \left|1-3\right|=2\ge 1$

So $1\cancel{R}3$

Thus $R$ in not transitive

Checking Anti symmetricity

If $1R2\Rightarrow \left|1-2\right|=1\le 1$

And $2R3\Rightarrow \left|3-2\right|=1\le 1$

But $1\ne 2$

Thus R us not anti symmetric

R is not transitive.

Hence, option A is the correct answer.