Relation Defined by \( b = a + 1 \) on Set \( A = \{1,2,3,4,5,6\} \)
📺 Video Explanation
📝 Question
Let \( A = \{1,2,3,4,5,6\} \). Define relation:
\[ R = \{(a,b) : b = a + 1\} \]
Check whether \( R \) is reflexive, symmetric, and transitive.
✅ Solution
🔹 Step 1: List the Relation
Possible pairs satisfying \( b = a + 1 \):
\[ R = \{(1,2), (2,3), (3,4), (4,5), (5,6)\} \]
🔹 Step 2: Reflexive
A relation is reflexive if: \[ (a,a) \in R \quad \forall a \in A \]
No pair like \( (1,1), (2,2), \dots \) is present.
❌ Therefore, the relation is Not Reflexive.
🔹 Step 3: Symmetric
A relation is symmetric if: \[ (a,b) \in R \Rightarrow (b,a) \in R \]
Example: \[ (1,2) \in R \text{ but } (2,1) \notin R \]
❌ Therefore, the relation is Not Symmetric.
🔹 Step 4: Transitive
A relation is transitive if: \[ (a,b) \in R \text{ and } (b,c) \in R \Rightarrow (a,c) \in R \]
Example: \[ (1,2), (2,3) \in R \Rightarrow (1,3) \text{ should be in } R \]
But: \[ (1,3) \notin R \]
❌ Therefore, the relation is Not Transitive.
🎯 Final Answer
✔ Reflexive: No
✔ Symmetric: No
✔ Transitive: No
\[ \therefore R \text{ is neither reflexive, nor symmetric, nor transitive} \]
🚀 Exam Insight
- Relations like \( b = a + 1 \) are directional (one-way).
- Such relations are usually not symmetric and not transitive.
- Always list pairs first for clarity.