### Sets And Relations Question 1

#### Question 1 - 24 January - Shift 1

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

(1) transitive but not reflexive

(2) symmetric but not transitive

(3) reflexive but not symmetric

(4) neither symmetric nor transitive

## Show Answer

#### Answer: (4)

#### Solution:

#### Formula: Reflexive relation (iv), Symmetric relation (v), Transitive relation (vi)

Reflexive : $(a, a) \Rightarrow \text{gcd}(a, a)=1$

Which is not true for every a $\epsilon$ Z.

Symmetric:

Take $a=2, b=1 \Rightarrow \text{gcd}(2,1)=1$

Also $2 a=4 \neq b$

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

Also now $2 a=2=b$

Hence $a=2 b$

$\Rightarrow R$ is not Symmetric

Transitive:

Let $a=14, b=19, c=21$

$gcd(a, b)=1$

$gcd(b, c)=1$

$gcd(a, c)=7$

Hence not transitive

$\Rightarrow R$ is neither symmetric nor transitive.