The question asks to determine the properties of the "is a brother of" relation within the set of children in a family. Let's break down the possible options:

Reflexive:

A relation is reflexive if every element is related to itself. In this case, the relation "is a brother of" would imply that every child is a brother of themselves. However, no one is typically considered their own brother. Therefore, the relation is not reflexive.

Symmetric:

A relation is symmetric if whenever a child $A$ is a brother of child $B$, child $B$ is also a brother of child $A$. Since being a brother is naturally symmetric, this condition holds. Hence, the relation is symmetric.

Transitive:

A relation is transitive if whenever a child $A$ is a brother of child $B$, and $B$ is a brother of child $C$, then $A$ must also be a brother of child $C$. While transitivity works in some contexts, being a brother does not necessarily imply that brothers of brothers are brothers, especially when there are different genders involved (e.g., if $C$ is a sister, $A$ and $C$ are not brothers). Hence, the relation is not transitive.

Conclusion:

The relation "is a brother of" is symmetric, but neither reflexive nor transitive.

The correct answer is:

B) Symmetric

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Would you like more details on why it’s not reflexive or transitive? Here are some related questions you might find interesting:

1. What is the difference between symmetric, reflexive, and transitive relations?
2. Can a relation be both symmetric and transitive without being reflexive?
3. How do different family structures affect the properties of relations like "is a sibling of"?
4. What other real-world examples have symmetric but not transitive relations?
5. How would the properties change if we considered the relation "is a sibling of" instead?

Tip: In set theory, it's crucial to precisely understand the nature of relations before determining their properties. Always analyze examples to clarify if a relation meets the required criteria.