Question

Mark the correct alternative in the following question:

Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is 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

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ R=\left\{\left(a,b\right):a\mathrm{is}\mathrm{brother}\mathrm{of}b\right\} \\ \mathrm{Let}\left(a,b\right)\in R.\mathrm{Then}, \\ a\mathrm{is}\mathrm{brother}\mathrm{of}b \\ \mathrm{but}b\mathrm{is}\mathrm{not}\mathrm{necessary}\mathrm{brother}\mathrm{of}a\left(\mathrm{As},b\mathrm{can}\mathrm{be}\mathrm{sister}\mathrm{of}a\right) \\ \Rightarrow \left(b,a\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{symmetric} \\ \mathrm{Also}, \\ \mathrm{Let}\left(a,b\right)\in R\mathrm{and}\left(b,c\right)\in R \\ \Rightarrow a\mathrm{is}\mathrm{brother}\mathrm{of}b\mathrm{and}b\mathrm{is}\mathrm{brother}\mathrm{of}c \\ \Rightarrow a\mathrm{is}\mathrm{brother}\mathrm{of}c \\ \Rightarrow \left(a,c\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{transitive} \end{gathered}$$

Hence, the correct alternative is option (b).