Relation \( R=\{(b,c)\} \) on Set \( A=\{a,b,c\} \)
📺 Video Explanation
📝 Question
Let \[ A=\{a,b,c\} \] and relation \[ R=\{(b,c)\} \] on \( A \).
Then \( R \) is:
- (a) reflexive only
- (b) symmetric only
- (c) transitive only
- (d) reflexive and transitive only
✅ Solution
We check reflexive, symmetric, and transitive properties.
🔹 Reflexive
For reflexive relation, all pairs \[ (a,a),(b,b),(c,c) \] must be in \( R \).
But:
\[ R=\{(b,c)\} \]
None of these self-pairs are present.
❌ Not reflexive.
🔹 Symmetric
If \[ (b,c)\in R \] then for symmetry:
\[ (c,b)\in R \] must also be present.
But:
\[ (c,b)\notin R \]
❌ Not symmetric.
🔹 Transitive
Transitive means:
If \[ (x,y)\in R \quad \text{and} \quad (y,z)\in R \] then \[ (x,z)\in R \]
Here relation has only one pair:
\[ (b,c) \]
There is no pair starting with \( c \), so no chain exists.
Thus transitivity condition is never violated.
✔ Relation is transitive.
🎯 Final Answer
\[ \boxed{\text{R is transitive only}} \]
✔ Correct option: (c)
🚀 Exam Shortcut
- Single pair relations are often transitive by default
- Reflexive needs all self-pairs
- Symmetric needs reverse pair