Question

The relation $R =\{(a, b)$ : $\operatorname{gcd}(a, b)=1,2 a \neq b, a, b \in Z\}$ is :

• A. reflexive but not symmetric
• B. transitive but not reflexive
• C. symmetric but not transitive
• D. neither symmetric nor transitive

Answer 1

The Correct Option is D

Reflexive: 
\((a,a) ⇒ gcd \space of (a,a)=1\)
Which is not true for every \(a ϵ Z.\)

Symmetric:
Take \(a=2, b=1⇒gcd(2,1)=1\)
Also \(2a=4\neq b\) 
Now when  \(a=1,b=2⇒gcd(1,2)=1\)
Also now  \(2a=2=b\) 
Hence \(a=2b\)
\(⇒\) R is not Symmetric

Transitive: 
Let \(a=14,b=19,c=21\)
gcd \((a,b)=1\)
gcd \((b,c)=1\)
gcd \((a,c)=7\)
Hence not transitive

\(⇒\)The correct option is (D): R is neither symmetric nor transitive.

Answer 2

Reflexive: For \( (a, a) \): \[ \gcd(a, a) = a, \] which is not equal to 1 for all \( a \). Hence, \( R \) is not reflexive. \vspace{0.2cm}
Symmetric: If \( (a, b) \in R \), then \( \gcd(a, b) = 1 \) and \( 2a \leq b \). However, for \( (b, a) \): \[ \gcd(b, a) = 1 \text{ is true, but } 2b \leq a \text{ may not hold}. \] Hence, \( R \) is not symmetric. \vspace{0.2cm} 
Transitive: Consider \( a = 14, b = 19, c = 21 \): \[ \gcd(14, 19) = 1, \quad \gcd(19, 21) = 1, \quad \gcd(14, 21) = 7. \] Thus, \( R \) is not transitive. 
Conclusion: \( R \) is neither symmetric nor transitive.