Question

Let A be the set of the children in a family. The relation ‘x’ is a brother of ‘y’ on A is
A. Reflexive
B. Symmetric
C. Transitive
D. None of these

Answer 1

Hint: In this question, we need to determine whether the given relation is reflexive, symmetric, and transitive or none of these. For this, we need to use the concept of relation.

Complete step-by-step answer:
We know that A is the set of the children in a family.
Also, we have been given that ‘x’ is a brother of ‘y’ on A.
Let R be the relation such as ‘is brother of’
Suppose \[x \in A\]
If x is a girl then x is not a brother of x.
Hence, \[\left( {x,x} \right) \notin R\]
So, R is not reflexive.
Now suppose that \[\left( {x,y} \right) \in R\]
That means, x is a brother of y.
That indicates y may or may not be a boy.
So, it is impossible that \[\left( {y,x} \right) \in R\]
Hence, R is not symmetric.
Let \[\left( {x,y} \right) \in R\] and \[\left( {y,z} \right) \in R\]
This indicates that ‘x is a brother of y and y is a brother of z’.
So, \[\left( {x,z} \right) \in R\]
So, R is clearly transitive.
Therefore, the correct option is (C).

Additional Information: We can say that the transitive relations are binary relations specified on a set so that if the first component is linked to the component and the component is connected to the set's third element, then the first component must be associated to the third component.

Note: Many students make mistakes in understanding the concept of relation. Sometimes they get confused with reflexive and transitive relations.