Relation: People Working at the Same Place
📺 Video Explanation
📝 Question
Let \( A \) be the set of all human beings in a town at a particular time. Define relation \( R = \{(x, y) : x \text{ and } y \text{ work at the same place}\} \). Determine whether \( R \) is reflexive, symmetric, and transitive.
✅ Solution
Step 1: Reflexive
A relation is reflexive if \( (x, x) \in R \) for all \( x \in A \).
Every person works at the same place as themselves.
\[ (x, x) \in R \quad \forall x \in A \]
✔ Therefore, the relation is Reflexive.
Step 2: Symmetric
A relation is symmetric if: \[ (x, y) \in R \Rightarrow (y, x) \in R \]
If person \( x \) works at the same place as \( y \), then \( y \) also works at the same place as \( x \).
✔ Therefore, the relation is 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 \( x \) and \( y \) work at the same place, and \( y \) and \( z \) also work at the same place, then \( x \) and \( z \) must work at that same place.
✔ Therefore, the relation is Transitive.
🎯 Final Conclusion
✔ Reflexive: Yes
✔ Symmetric: Yes
✔ Transitive: Yes
\[ \therefore R \text{ is an Equivalence Relation} \]
🚀 Exam Insight
- Relations like “same school”, “same class”, “same workplace” are always equivalence relations.
- They form groups called equivalence classes.
- Think: common property → always R, S, T