Question

Consider the non - empty set consisting of children in a family and a relation R defined as $aRb$ if a is brother of b. Then R is ?

• A. transitive but not symmetric
• B. symmetric but not transitive
• C. both symmetric and transitive
• D. neither symmetric nor transitive

Answer 1

The correct option is A transitive but not symmetric

Check whether the given relation is symmetric

Given: $R=\{aRb,a\text{is brother of}b\},\forall ab\in R$

Let $aRb\Rightarrow a$ is brother of $b$

Then $bRa\Rightarrow b$ is brother of $a$ is not true

because $b$ may be sister of $a.$

$\therefore aRb\nRightarrow bRa$

So, R is not symmetric

Check whether the given relation is transitive

Let $aRb$ and $bRc$

$\Rightarrow aRb\Rightarrow a$ is brother of b

$\&\Rightarrow bRc\Rightarrow b$ is brother of c

Hence, $a$ is brother of c.

$\therefore aRb\text{and}bRc\Rightarrow aRc.$

$\therefore R$ is transitive

Hence, R is transitive but not symmetric

$\therefore$ Option (B) is correct.