$\begin{array}{1 1}(A) \text{reflexive but not symmetric}\\\text{ (B) reflexive but not Transitive} \\ \text{ (C) symmetric and transitive} \\ \text{(D) neither symmetric, nor transitive} \end{array}$

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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 $R=\{(1,1),(2,2),(3,3),(1,2),(2,3)(1,3)\}$

$(1,1)(2,2)(3,3) \in R$

Hence $(a,a) \in$ R for $a \in \{1,2,3\}$

R is reflexive

$(1,2) \in R$ but $(2,1) \notin R$

R is not symmetric

$(1,2) \in R (2,3) \in R$ also $(1,3) \in R$

R is transitive

R is reflexive, transitive but not symmetric

Solution: 'A' option is correct

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