Relation Defined by \( a^2 – 4ab + 3b^2 = 0 \) on \( \mathbb{R} \)
📺 Video Explanation
📝 Question
Let \( R_3 \) be a relation on \( \mathbb{R} \) defined by:
\[ (a, b) \in R_3 \iff a^2 – 4ab + 3b^2 = 0 \]
Test whether \( R_3 \) is reflexive, symmetric, and transitive.
✅ Solution
🔹 Step 1: Simplify the Relation
\[ a^2 – 4ab + 3b^2 = (a – b)(a – 3b) \]
So, \[ (a, b) \in R_3 \iff a = b \quad \text{or} \quad a = 3b \]
🔹 Step 2: Reflexive
A relation is reflexive if: \[ (a, a) \in R_3 \quad \forall a \in \mathbb{R} \]
Substitute: \[ a^2 – 4a^2 + 3a^2 = 0 \]
\[ 0 = 0 \]
✔ True for all \( a \)
✔ Therefore, the relation is Reflexive.
🔹 Step 3: Symmetric
If \( (a, b) \in R_3 \), then: \[ a = b \quad \text{or} \quad a = 3b \]
Case 1: \( a = b \Rightarrow b = a \) ✔
Case 2: \( a = 3b \Rightarrow b = \frac{a}{3} \)
Now check: \[ b^2 – 4ab + 3a^2 = ? \]
Not necessarily zero.
❌ Therefore, the relation is Not Symmetric.
🔹 Step 4: Transitive
If: \[ (a, b) \in R_3 \Rightarrow a = b \text{ or } a = 3b \] \[ (b, c) \in R_3 \Rightarrow b = c \text{ or } b = 3c \]
Check cases:
- If \( a = b \) and \( b = c \) ⇒ \( a = c \) ✔
- If \( a = 3b \) and \( b = 3c \) ⇒ \( a = 9c \) ❌ (not in relation)
So, transitivity fails.
❌ Therefore, the relation is Not Transitive.
🎯 Final Answer
✔ Reflexive: Yes
✔ Symmetric: No
✔ Transitive: No
\[ \therefore R_3 \text{ is reflexive only} \]
🚀 Exam Insight
- Always factor quadratic relations first.
- Convert into simple forms like \( a = b \) or \( a = kb \).
- Test symmetry and transitivity using cases.