Question

In $ aRb $ if “ $ a $ and $ b $ are animals in different zoological parks” then $ R $ is:
A.Only reflexive
B.Only symmetric
C.Only transitive
D.Equivalence

Answer 1

Hint: Check the definitions of the type of relations for the given relation and check whether the given relation is reflexive, symmetric or transitive or all of them i.e. an equivalence relation.

Complete step-by-step answer:
As given in the question $ aRb $ if “ $ a $ and $ b $ are animals in different zoological parks”.
For a relation to be reflexive, $ \left( {a,a} \right) $ must belong to the relation set. If the element $ \left( {a,a} \right) $ belongs to the relation set this means that $ a $ and $ a $ are from different zoological parks. This is not possible at all.
Now for a relation to be symmetric, if $ \left( {a,b} \right) $ belongs to the relation set then $ \left( {b,a} \right) $ must also belong to the relation set.
If $ \left( {a,b} \right) $ belongs to the relation set, this means that $ a $ and $ b $ are from different zoological parks and if $ \left( {b,a} \right) $ belongs to the relation set, this means that $ b $ and $ a $ are from different zoological parks.
This is always true. This means that the given relation is symmetric.
Now for a relation to be transitive, if $ \left( {a,b} \right) $ and $ \left( {b,c} \right) $ belongs to the relation set then $ \left( {a,c} \right) $ must also belong to the relation set.
If $ \left( {a,b} \right) $ belongs to the relation set, this means that $ a $ and $ b $ are from different zoological parks and if $ \left( {b,c} \right) $ belongs to the relation set, this means that $ b $ and $ c $ are from different zoological parks.
It is not always true that $ a $ and $ c $ are from different zoological parks as they both can be from the same zoological park also.
So, the given relation is not a transitive relation.
So, the given relation is only a symmetric relation.
So, the correct answer is “Option C”.

Note: Please kindly take a look at the case for the transitivity of relation when $ a $ and $ c $ are from the same zoological park and $ b $ is from different zoological parks. Then $ \left( {a,b} \right) $ belongs to the relation set and also $ \left( {b,c} \right) $ belongs to the relation set but $ \left( {a,c} \right) $ doesn’t belong to relation set. So, the given relation is not transitive.