$\begin{array}{1 1}(a)\;symmetric\;but\;not\;transitive & (b)\;transitive\;but\;not\;symmetric\\(c)\;neither\;symmetric\;nor\;transitive & (d)\;both\;symmetric\;and\; transitive\end{array}$

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- A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for every $\; a\in\;A$
- A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
- A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$

Given:Non empty set containing children in a family $ a R b$ a is brother of b

but b need not be brother of a ,can also be sister of a

Therefore it does not imply b R a

R is not symmetric

$ a R b,b R c$

a is brother of b and b is brother of c

Therefore a must be brother of c

=>$aRc$

Therefore R is transitive

Relation R is transitive. but not symmetric

Solution:'B' option is correct

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