Question

Let A be the non-void 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

Given that,

A is the non-void set of the children in a family.

To find,

Out of the four options, The relation 'x is a brother of y' on A is?

Solution,

Reflexive :

R is reflexive, $\forall x\ \epsilon\ A, \ xRx$

Symmetric :

R is symmetric, $\forall x,\ y\ \epsilon\ A, \ xRy\rightarrow yRx$

Transitive :

R is transitive, $\forall x,y,z\ \epsilon\ A, \ xRy \ \text{and} \ yRz \rightarrow xRz$

In this case, A is a set of children in a family.

aRb means a is brother of b. But b should not be bother of a. It should be only sister of a. So, R is not symmetric.

a is brother of b and b is brother of c. It means that a must be brother of c i.e. aRc.

Hence, the correct option is (c) "transitive".