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

Answer: (D)

Reflexive: $( a , a ) \Rightarrow \operatorname{gcd}$ of $( a , a )=1$
Which is not true for every a $\epsilon Z$.
Symmetric:
Take $a =2, b =1 \Rightarrow \operatorname{gcd}(2,1)=1$
Also $2 a =4 \neq b$
Now when $a =1, b =2 \Rightarrow \operatorname{gcd}(1,2)=1$
Also now $2 a =2= b$
Hence $a=2 b$
$\Rightarrow R$ 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.