The relation [R = (a,b): (a,b) = 1,2a b,a,b Z] is :

Question

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

Accepted Answer

Given,

(a, b) belongs to relation R if $$\gcd (a,b) = 1, 2a \ne b$$.

Here $$\gcd $$ means greatest common divisor. $$\gcd $$ of two numbers is the largest number that divides both of them.

(1) For Reflexive,

In $$aRa,\,\gcd (a,a) = a$$

$$\therefore$$ This relation is not reflexive.
(2) For Symmetric:

Take $a=2, b=1 \Rightarrow \operatorname{gcd}(2,1)=1$
Also $2 a=4 \neq b$

Now $$\gcd (b,a) = 1$$ $ \Rightarrow \operatorname{gcd}(1,2)=1$

and 2b should not be equal to a

But here, $2 b=2=a$

$\Rightarrow \mathrm{R}$ is not Symmetric

(3) For Transitive:

Let $\mathrm{a}=14, \mathrm{~b}=19, \mathrm{c}=21$

$\operatorname{gcd}(\mathrm{a}, \mathrm{b})=1, 2a \ne b$

$\operatorname{gcd}(\mathrm{b}, \mathrm{c})=1, 2b \ne c$

$\operatorname{gcd}(\mathrm{a}, \mathrm{c})=7, 2a \ne c$

Hence not transitive

$\Rightarrow R$ is neither symmetric nor transitive.

Solutions · 1