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)

neither symmetric nor transitive

Answer 2

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.