(c) : If \textquotesingle{} $R$ \textquotesingle{} be a relation defined on $N$ as $aRb$ is $2a + 3b$ is a multiple of $5,a,b \in \ N$.\\ (i) $aRa \Rightarrow 5a$ is a multiple of 5\\ $R$ is a reflexive relation,\\ (ii) $aRb,2a + 3b = 5\alpha\$ (let) Now, $bRA \Rightarrow 2b + 3a = 2b + \left( \frac{5\alpha - 3b}{2} \right) \cdot 3$\\ $= \frac{15}{2}\alpha - \frac{5}{2}b = \frac{5}{2}(3\alpha - b) = \frac{5}{2}(2a + 2b - 2\alpha) = 5(a + b - \alpha)$ So, $R$ is symmetric relation.\\ (iii) $aRb \Rightarrow 2a + 3b = 5\alpha;bRc \Rightarrow 2b + 3c = 5\beta$ Now, $2a + 5b + 3c = 5(\alpha + \beta)$\\ $\Rightarrow 2a + 5b + 3c = 5(\alpha + \beta)$ or $2a + 3c = 5(\alpha + \beta - b)$\\ $\Rightarrow aRc$ So, $R$ is a transitive relation.\\ Hence relation \textquotesingle{} $R$ \textquotesingle{} is an equivalence relation. \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \setcounter{enumi}{12}