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

The correct option is "transitive but not symmetric".

Let us analyze the given relation $R$ defined on a non-empty set of children in a family.
The relation is defined as: $aRb$ if and only if $a$ is the brother of $b$.

1. Checking for Symmetry:
A relation $R$ is symmetric if $aRb$ implies $bRa$ for all elements in the set.
Suppose $aRb$ is true. This means $a$ is a brother of $b$. Since $a$ is a brother, $a$ must be male.
However, $b$ can be either male (brother) or female (sister) since they belong to the same family.
If $b$ is female, then $b$ is the sister of $a$, which means $b$ is not a brother of $a$. Therefore, $bRa$ is not true.
Thus, the relation $R$ is not symmetric.

2. Checking for Transitivity:
A relation $R$ is transitive if $aRb$ and $bRc$ imply $aRc$ for all elements in the set.
Suppose $aRb$ and $bRc$ are true.
This means:
- $a$ is the brother of $b$ (which implies $a$ is male and shares the same parents as $b$).
- $b$ is the brother of $c$ (which implies $b$ is male and shares the same parents as $c$).
Since $a$, $b$, and $c$ all share the same parents, and $a$ is male, it follows that $a$ must be the brother of $c$.
Therefore, $aRc$ is true.
Thus, the relation $R$ is transitive.

Conclusion: The relation $R$ is transitive but not symmetric.