Relation \( \geq \) on Real Numbers
📺 Video Explanation
📝 Question
Show that the relation \( R \) defined on \( \mathbb{R} \) by:
\[ (a, b) \in R \iff a \geq b \]
is reflexive and transitive but not symmetric.
✅ Solution
🔹 Step 1: Reflexive
A relation is reflexive if: \[ (a, a) \in R \quad \forall a \in \mathbb{R} \]
Since: \[ a \geq a \]
✔ True for all real numbers
✔ Therefore, the relation is Reflexive.
🔹 Step 2: Symmetric
A relation is symmetric if: \[ (a, b) \in R \Rightarrow (b, a) \in R \]
If: \[ a \geq b \]
This does not imply: \[ b \geq a \]
Example: \[ 5 \geq 3 \ (\text{true}),\quad 3 \geq 5 \ (\text{false}) \]
❌ Therefore, the relation is Not Symmetric.
🔹 Step 3: Transitive
A relation is transitive if: \[ (a, b) \in R \text{ and } (b, c) \in R \Rightarrow (a, c) \in R \]
If: \[ a \geq b \text{ and } b \geq c \]
Then: \[ a \geq c \]
✔ Therefore, the relation is Transitive.
🎯 Final Conclusion
✔ Reflexive: Yes
❌ Symmetric: No
✔ Transitive: Yes
\[ \therefore \text{Relation } \geq \text{ is reflexive and transitive but not symmetric} \]
🚀 Exam Insight
- All order relations (\( >, <, \geq, \leq \)) are transitive
- They are reflexive if equality is included (≥, ≤)
- They are never symmetric