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 gcd(a, a)=1
Which is not true for every a $\epsilon$ Z.
Symmetric:
Take $a=2, b=1 \Rightarrow gcd(2,1)=1$
Also $2 a=4 \neq b$
Now when $a=1, b=2 \Rightarrow gcd(1,2)=1$
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.
