Let R1 and R2 be two relations defined as follows: R1=(a,b)∈R2:a2+b2∈Q and R2=(a,b)∈R2:a2+b2∉Q , where Q is the set of all rational numbers, then

The given relation is R1=(a,b)∈R2:a2+b2∈Q and R2=(a,b)∈R2:a2+b2∉Q For the relation R1:a=1+2,b=1-2,c=234 It gives (a,b)∈R1,(b,c)∈R2 but (a,c)∉R Hence, the relation R1=(a,b)∈R2:a2+b2∈Q is not transitive  Consider the relation R2=(a,b)∈R2:a2+b2∉Q For the relation R2:a=1+2,b=1+22,c=1-2 It gives (a,b)∈R1,(b,c)∈R2 but (a,c)∉R Hence, the relation R2=(a,b)∈R2:a2+b2∉Q is not transitive