Question

Consider a non – empty set consisting of children in a family and a relation R defined as $aRb$, if a is a brother of b, then R is
(a) Symmetric but not transitive
(b) Transitive but not symmetric
(c) Neither symmetric nor transitive
(d) Both symmetric and transitive

Answer 1

Hint: To check if the relation is symmetric, considering the given fact that ‘a’ is a brother of ‘b’ check if ‘b’ is also the brother of ‘a’. If yes then the relation is symmetric otherwise not. Now, to check for the transitive nature of the relation, assume that ‘b’ is the brother of ‘c’ apart from the fact that ‘a’ is the brother of ‘b’, check if ‘a’ is also the brother of ‘c’. If yes then the relation is transitive otherwise not.

Complete step by step answer:
Here we have been provided with a relation R such that ‘a’ is a brother of ‘b’. We have to determine the symmetric and the transitive nature of the relation. Let us check them one by one.
(1) A relation is symmetric in nature when we have $\left( a,b \right)\in R$ then $\left( b,a \right)\in R$. In the question it is said that ‘a’ is a brother of ‘b’ that means the relation will be symmetric if and only if ‘b’ is also the brother of ‘a’.
Now, it may be possible that b is a sister of ‘a’ in the same family therefore ‘b’ may not be related with ‘a’ in the similar relation as ‘a’ is related with ‘b’. Therefore the relation is not symmetric.
(2) A relation is transitive in nature when we have $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$ then $\left( a,c \right)\in R$. It is given that ‘a’ is a brother of ‘b’ so we assume that there is a third child such that ‘b’ is a brother of ‘c’.
Now, it is obvious that if ‘a’ is a brother of ‘b’ and ‘b’ is a brother of ‘c’ then ‘a’ will also be the brother of ‘c’. Therefore the relation is transitive.

So, the correct answer is “Option b”.

Note: Note that there is one more relation known as the reflexive relation. In that relation we have to check if ‘a’ is the brother of ‘a’. It is not possible that ‘a’ will be a brother of itself, so the relation is not reflexive. If a relation is transitive, symmetric and reflexive then it is called the equivalence relation.