Let
be the set of the children in a family. The relation
is a brother of
on

is

A

Reflexive
B

Symmetric
C

Transitive
D

None of these

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.

Let the set of the children in a family. The relation 'is a brother of' is: A) Reflexive B) Symmetric C) Transitive D) None of these

Explore the properties of the 'is a brother of' relation in set theory. Learn why this relation is symmetric but not reflexive or transitive, with a breakdown of key concepts like reflexivity, symmetry, and transitivity. Ideal for high school students studying relations in set theory.

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