Question

The following relation is defined on the set of real numbers. a R b iff $|a-b|>0.$Choose the correct options ?

• A. It is reflexive
• B. It is transitive
• C. It is symmetric
• D. none of these

Answer 1

Answer: (C)

It is symmetric

Given definition of

$aRb$ $iff|a-b|>0.$
Reflexivity:

R is not reflexive since $|a-a|=0$ and so $|a-a|\ngtr 0$

Thus $a(\sim R)a$ for any real number a.
Symmetry:

R is symmetric since if $|a-b|>0,$ then

$|b-a|=|a-b|>0.$

Thus $aRb$

$\Rightarrow bRa$
Transitivity:

$R$ is not transitive.

For example: Consider the numbers $3,7,3.$

Then we have $3R7$ since $|3-7|=4>0$ and $7R3$

since $|7-3|=4>0.$

But $3(\sim R)3since|3-3|=0$