QJEEM23Ma7472: JEE Main 2023 Mathematics - Class 11 JEEM23Ma7472 Solution: The relation R = { ( a , b ) : gcd ( a , b ) = 1 , 2 a ≠ b , a…

Question

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

Accepted Answer

Given,

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

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

(1) For Reflexive,

In $a R a, gcd (a, a) = a$

$∴$ 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.

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