Relation: “Father of” Relation
📺 Video Explanation
📝 Question
Let \( A \) be the set of all human beings in a town at a particular time. Define relation \( R = \{(x, y) : x \text{ is father of } y\} \). Determine whether \( R \) is reflexive, symmetric, and transitive.
✅ Solution
Step 1: Reflexive
A relation is reflexive if \( (x, x) \in R \) for all \( x \in A \).
This would mean a person is father of themselves, which is not possible.
\[ (x, x) \notin R \]
❌ Therefore, the relation is Not Reflexive.
Step 2: Symmetric
A relation is symmetric if: \[ (x, y) \in R \Rightarrow (y, x) \in R \]
If \( x \) is father of \( y \), then \( y \) is child of \( x \), not father.
\[ (x, y) \in R \nRightarrow (y, x) \in R \]
❌ Therefore, the relation is Not Symmetric.
Step 3: Transitive
A relation is transitive if: \[ (x, y) \in R \text{ and } (y, z) \in R \Rightarrow (x, z) \in R \]
If \( x \) is father of \( y \), and \( y \) is father of \( z \), then \( x \) is grandfather of \( z \), not father.
\[ (x, z) \notin R \]
❌ Therefore, the relation is Not Transitive.
🎯 Final Conclusion
✔ Reflexive: No
✔ Symmetric: No
✔ Transitive: No
\[ \therefore R \text{ is neither reflexive, nor symmetric, nor transitive} \]
🚀 Exam Insight
- Relations like “father of”, “mother of”, “wife of” are directional.
- They fail all three properties: reflexive, symmetric, and transitive.
- Hence, they are not equivalence relations.