JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023The relation R={(a,b):gcd(a,b)=1,2a=b,a,b∈Z} is:
Choose the correct answer:
- A.
reflexive but not symmetric
- B.
transitive but not reflexive
- C.
symmetric but not transitive
- D.
neither symmetric nor transitive
(Correct Answer)
neither symmetric nor transitive
Explanation
Solution for Relation R:
R={(a,b):gcd(a,b)=1,2a=b,a,b∈Z}
Reflexive:
Check for (a,a)
gcd(a,a)=a
So, R is not reflexive.
Symmetric:
(a,b)∈R⟹gcd(a,b)=1
Now check for (b,a)
gcd(b,a)=gcd(a,b)=1
But gcd(1,2)=gcd(2,1)=1
But b=2a, So R is not symmetric.
Transitive:
Consider, gcd(2,3)=1
⟹(2,3)∈R
Now, gcd(3,4)=1
⟹(3,4)∈R
Again, gcd(2,4)=2=1
⟹(2,4)∈/R
⟹R is not Transitive.
Explanation
Solution for Relation R:
R={(a,b):gcd(a,b)=1,2a=b,a,b∈Z}
Reflexive:
Check for (a,a)
gcd(a,a)=a
So, R is not reflexive.
Symmetric:
(a,b)∈R⟹gcd(a,b)=1
Now check for (b,a)
gcd(b,a)=gcd(a,b)=1
But gcd(1,2)=gcd(2,1)=1
But b=2a, So R is not symmetric.
Transitive:
Consider, gcd(2,3)=1
⟹(2,3)∈R
Now, gcd(3,4)=1
⟹(3,4)∈R
Again, gcd(2,4)=2=1
⟹(2,4)∈/R
⟹R is not Transitive.