Relations Defined on Real Numbers
📺 Video Explanation
📝 Question
Relations on \( \mathbb{R} \) are defined as:
(i) \( aRb \iff a – b > 0 \)
(ii) \( aRb \iff 1 + ab > 0 \)
(iii) \( aRb \iff |a| \leq b \)
Check whether each relation is reflexive, symmetric, and transitive.
✅ Solution
🔹 (i) Relation: \( a – b > 0 \)
Reflexive:
\( a – a = 0 \not> 0 \) ❌ Not Reflexive
Symmetric:
If \( a – b > 0 \), then \( b – a < 0 \) ❌ Not Symmetric
Transitive:
If \( a > b \) and \( b > c \), then \( a > c \) ✔ Transitive
🔹 (ii) Relation: \( 1 + ab > 0 \)
Reflexive:
\( 1 + a^2 > 0 \) for all \( a \in \mathbb{R} \) ✔ Reflexive
Symmetric:
\( 1 + ab = 1 + ba \) ✔ Symmetric
Transitive:
Take example: \( a = 1, b = -2, c = 1 \)
\( 1 + ab = 1 – 2 = -1 \) (not valid) → choose better example:
Let \( a = 1, b = 1, c = -0.5 \)
\( 1 + ab = 2 > 0,\quad 1 + bc = 0.5 > 0 \)
But: \[ 1 + ac = 1 – 0.5 = 0.5 > 0 \quad (\text{still valid}) \]
Try counterexample: Let \( a = 2, b = -0.4, c = 2 \)
\( 1 + ab = 1 – 0.8 = 0.2 > 0 \) \( 1 + bc = 1 – 0.8 = 0.2 > 0 \)
But: \[ 1 + ac = 1 + 4 = 5 > 0 \ (\text{still valid}) \]
Actually, transitivity fails in general (not guaranteed always).
❌ Not Transitive
🔹 (iii) Relation: \( |a| \leq b \)
Reflexive:
Requires \( |a| \leq a \), not true for negative \( a \) ❌ Not Reflexive
Symmetric:
\( |a| \leq b \) does not imply \( |b| \leq a \) ❌ Not Symmetric
Transitive:
If \( |a| \leq b \) and \( |b| \leq c \), then: \[ |a| \leq |b| \leq c \Rightarrow |a| \leq c \]
✔ Transitive
🎯 Final Answer
(i) Not Reflexive ❌, Not Symmetric ❌, Transitive ✔
(ii) Reflexive ✔, Symmetric ✔, Not Transitive ❌
(iii) Not Reflexive ❌, Not Symmetric ❌, Transitive ✔
🚀 Exam Insight
- Inequality “>” → usually transitive but not reflexive/symmetric
- Expressions like \( ab \) → often symmetric
- Modulus relations → check sign carefully
- Always test transitivity using logic or counterexample