Question:
A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set \(\mathbf{A = \{1, 2, 3\}}\):\[\begin{aligned}
R_1 &= \{(2, 3), (3, 2)\} \\
R_2 &= \{(1, 2), (1, 3), (3, 2)\} \\
R_3 &= \{(1, 2), (2, 1), (1, 1)\} \\
R_4 &= \{(1, 1), (1, 2), (3, 3), (2, 2)\} \\
R_5 &= \{(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)\}
\end{aligned}\]The students are asked to answer the following questions about the above relations:(i) Identify the relation which is reflexive, transitive but not symmetric.(ii) Identify the relation which is reflexive and symmetric but not transitive.(iii) (a) Identify the relations which are symmetric but neither reflexive nor transitive.\[\mathbf{OR}\](iii) (b) What pairs should be added to the relation \(\mathbf{R_2}\) to make it an equivalence relation?