Brother Relation in a Family
📺 Video Explanation
📝 Question
Let a non-empty set consist of children in a family.
A relation \(R\) is defined by:
\[ aRb \iff a \text{ is brother of } b \]
Then, \(R\) is:
- A. symmetric but not transitive
- B. transitive but not symmetric
- C. neither symmetric nor transitive
- D. both symmetric and transitive
✅ Solution
🔹 Symmetric Check
If \(a\) is brother of \(b\), then:
- \(a\) is male
- \(a\) and \(b\) are siblings
But \(b\) may be sister of \(a\), not brother.
Example:
Rahul is brother of Riya, but Riya is not brother of Rahul.
❌ Not symmetric.
🔹 Transitive Check
If:
\[ a \text{ is brother of } b \quad \text{and} \quad b \text{ is brother of } c \]
Then:
- \(a,b,c\) are siblings in same family
- \(a\) is male
So:
\(a\) is brother of \(c\).
✔ Transitive.
🎯 Final Answer
\[ \boxed{\text{Transitive but not symmetric}} \]
✔ Correct option: B
🚀 Exam Shortcut
- Brother relation depends on gender
- Reverse may fail if sibling is female
- Sibling chain keeps brother relation valid