Question

If \(R\) be a relation defined as \(aRb\) iff \(|a-b|>0\), then the relation is

• A. reflexive
• B. transitive
• C. symmetric and transitive
• D. symmetric

Hint:

Relation \(a\neq b\) is symmetric but not reflexive and not transitive.

Answer 1

The Correct Option is D

Step 1: Interpret relation.
\[ aRb \iff |a-b|>0 \] 
This means: 
\[ aRb \iff a\neq b \] 
Step 2: Check reflexive property. 
Reflexive means \(aRa\) for all \(a\). 
But: 
\[ |a-a|=0 \not> 0 \] 
So it is not reflexive. 
Step 3: Check symmetric property. 
If \(aRb\), then \(a\neq b\). 
Then \(b\neq a\Rightarrow |b-a|>0\Rightarrow bRa\). 
So it is symmetric. 
Step 4: Check transitive property. 
If \(aRb\) and \(bRc\), then \(a\neq b\) and \(b\neq c\). 
But it is possible that \(a=c\). Example: \(a=1,b=2,c=1\). 
Then \(a=c\Rightarrow |a-c|=0\), so \(aRc\) is false. 
So it is not transitive. 
Final Answer: 
\[ \boxed{\text{Symmetric}} \]