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1.Relation and Function
hard
The relation $R =\{( a , b ): \operatorname{gcd}( a , b )=1,2 a \neq b , a , b \in Z \}$ is:
A
transitive but not reflexive
B
symmetric but not transitive
C
reflexive but not symmetric
D
neither symmetric nor transitive
(JEE MAIN-2023)
Solution
Reflexive : $(a, a) \Rightarrow \operatorname{gcd}$ of $(a, a)=1$
Which is not true for every a $\epsilon Z$.
Symmetric:
Take $a =2, b =1 \Rightarrow \operatorname{gcd}(2,1)=1$
Also $2 a=4 \neq b$
Now when $a =1, b =2 \Rightarrow \operatorname{gcd}(1,2)=1$
Also now $2 a =2= b$
Hence $a=2 b$
$\Rightarrow R$ is not Symmetric
Transitive:
Let $a =14, b =19, c =21$
$\operatorname{gcd}( a , b )=1$
$\operatorname{gcd}(b, c)=1$
$\operatorname{gcd}( a , c )=7$
Hence not transitive
$\Rightarrow R$ is neither symmetric nor transitive.
Standard 12
Mathematics
