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Solution


Q.

Let R be a relation on R, given by

R = {(a,b):3a-3b+7 is an irrational number}.

Then R is                                                         [2023]
1 reflexive and symmetric but not transitive  
2 reflexive and transitive but not symmetric  
3 reflexive but neither symmetric nor transitive  
4 an equivalence relation

Ans.

(3)

R={(a,b):3a-3b+7 is an irrational number}

Reflexive: For (a,a), we have 3a-3a+7=7, which is an irrational number. ∴R is reflexive.

Symmetric: Let (a,b)∈R, i.e., 3a-3b+7 is an irrational number.

Now, we need to check (b,a)∈R or not.

Let 3a=7 and 3b=8

Then 3a-3b+7=7-8+7=27-8, which is an irrational number.

But 3b-3a+7=8-7+7=8, which is not an irrational number.

∴ For (a,b)∈R, (b,a)∉R.

∴R is not symmetric.

Transitive: Let (a,b) and (b,c)∈R.

Let 3a=8, 3b=27, 3c=7

Then 3a-3b+7=8-27+7=8-7, which is an irrational number.

Also, 3b-3c+7=27-7+7=27, which is an irrational number.

But 3a-3c+7=8-7+7=8, which is not an irrational⇒(a,c)∉R.

∴  R is not transitive.