Question

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

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

Answer 1

Answer: (D)

Reflexive: $(a,a)\Rightarrow \gcd$ of $(a,a)=1$
Which is not true for every a $ϵZ$.
Symmetric:
Take $a=2,b=1\Rightarrow \gcd (2,1)=1$
Also $2a=4\ne b$
Now when $a=1,b=2\Rightarrow \gcd (1,2)=1$
Also now $2a=2=b$
Hence $a=2b$
$\Rightarrow R$ is not Symmetric
Transitive:
Let $a=14,b=19,c=21$
$\text{gcd}(a,b)=1$
$\text{gcd}(b,c)=1$
$\text{gcd}(a,c)=7$
Hence not transitive
$\Rightarrow R$ is neither symmetric nor transitive.