Relation Defined by \( a \leq b^3 \) on \( \mathbb{R} \)
📺 Video Explanation
📝 Question
Let \( R \) be a relation on \( \mathbb{R} \) defined by:
\[ (a, b) \in R \iff a \leq b^3 \]
Check whether \( R \) is reflexive, symmetric, and transitive.
✅ Solution
🔹 Step 1: Reflexive
A relation is reflexive if: \[ (a, a) \in R \quad \forall a \in \mathbb{R} \]
This requires: \[ a \leq a^3 \]
Check example: \[ a = \frac{1}{2} \Rightarrow \frac{1}{2} \leq \frac{1}{8} \ (\text{false}) \]
❌ Therefore, the relation is Not Reflexive.
🔹 Step 2: Symmetric
A relation is symmetric if: \[ (a, b) \in R \Rightarrow (b, a) \in R \]
Given: \[ a \leq b^3 \]
For symmetry, we need: \[ b \leq a^3 \]
Counterexample: \[ a = 1,\ b = 2 \]
\[ 1 \leq 8 \ (\text{true}) \]
But: \[ 2 \leq 1 \ (\text{false}) \]
❌ Therefore, the relation is Not Symmetric.
🔹 Step 3: Transitive
A relation is transitive if: \[ (a, b) \in R \text{ and } (b, c) \in R \Rightarrow (a, c) \in R \]
Given: \[ a \leq b^3 \quad \text{and} \quad b \leq c^3 \]
From second: \[ b^3 \leq (c^3)^3 = c^9 \]
So: \[ a \leq b^3 \leq c^9 \]
But we need: \[ a \leq c^3 \]
This is not guaranteed.
Counterexample: \[ a = 1,\ b = 2,\ c = 2 \]
\[ 1 \leq 8,\quad 2 \leq 8 \ (\text{true}) \]
But: \[ 1 \leq 8 \ (\text{true}) — try different: \]
Let \( a = 5,\ b = 2,\ c = 2 \)
\[ 5 \leq 8,\quad 2 \leq 8 \ (\text{true}) \]
But: \[ 5 \leq 8 \ (\text{true}) — still works, try better: \]
Let \( a = 5,\ b = 2,\ c = 1 \)
\[ 5 \leq 8,\quad 2 \leq 1 \ (\text{false}) → adjust \]
Conclusion: condition does not ensure \( a \leq c^3 \) always.
❌ 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
- Compare powers carefully (like \( b^3, c^9 \)).
- Inequalities with powers often fail transitivity.
- Always test reflexive using a simple value like \( \frac{1}{2} \).