1. **State the problem:** We have two sets: Set A (Professions) = {Lecturer, Pilot, Doctor, Musician} Set B (Tools/Objects) = {Stethoscope, Whitecoat, Musical Instrument, Book} (a)(i) Design a relation $R$ from Set A to Set B where $aRb$ means "Profession $a$ uses or requires tool $b$." The relation should have at least five elements. Given relation matrix: - Lecturer relates to Musical Instrument and Book - Pilot relates to Book - Doctor relates to Stethoscope and Book - Musician relates to Whitecoat 2. **Describe relation $R$ in words:** The relation $R$ pairs each profession with the tools they use or require. For example, a Lecturer uses a Musical Instrument and a Book, a Doctor uses a Stethoscope and a Book, etc. 3. **Specify relation $R$ elements using ordered pairs:** $$R = \{(\text{Lecturer}, \text{Musical Instrument}), (\text{Lecturer}, \text{Book}), (\text{Pilot}, \text{Book}), (\text{Doctor}, \text{Stethoscope}), (\text{Doctor}, \text{Book}), (\text{Musician}, \text{Whitecoat})\}$$ 4. **(a)(ii) Write $R$ in infix notation and $R^{-1}$ in ordered pair notation:** - Infix notation: $aRb$ means "Profession $a$ uses or requires tool $b$." - Inverse relation $R^{-1}$ swaps each pair: $$R^{-1} = \{(\text{Musical Instrument}, \text{Lecturer}), (\text{Book}, \text{Lecturer}), (\text{Book}, \text{Pilot}), (\text{Stethoscope}, \text{Doctor}), (\text{Book}, \text{Doctor}), (\text{Whitecoat}, \text{Musician})\}$$ 5. **(a)(iii) Is $R$ a function?** A relation is a function if every element in the domain (Set A) relates to exactly one element in the codomain (Set B). - Lecturer relates to two tools (Musical Instrument, Book) - Doctor relates to two tools (Stethoscope, Book) - Pilot and Musician relate to one tool each Since Lecturer and Doctor relate to more than one tool, $R$ is **not** a function. 6. **(b) Design relation $S$ from Set B to itself with at least five elements, reflexive and symmetric but not transitive.** Example relation $S$: $$S = \{(\text{Stethoscope}, \text{Stethoscope}), (\text{Whitecoat}, \text{Whitecoat}), (\text{Musical Instrument}, \text{Musical Instrument}), (\text{Book}, \text{Book}), (\text{Stethoscope}, \text{Whitecoat}), (\text{Whitecoat}, \text{Stethoscope})\}$$ - Reflexive: Every element relates to itself (all $(b,b)$ pairs present). - Symmetric: If $(b_1,b_2)$ is in $S$, then $(b_2,b_1)$ is also in $S$ (e.g., $(\text{Stethoscope}, \text{Whitecoat})$ and $(\text{Whitecoat}, \text{Stethoscope})$). - Not transitive: For example, $(\text{Stethoscope}, \text{Whitecoat})$ and $(\text{Whitecoat}, \text{Stethoscope})$ are in $S$, but $(\text{Stethoscope}, \text{Stethoscope})$ is in $S$ (reflexive), but if we add $(\text{Whitecoat}, \text{Book})$ without $(\text{Stethoscope}, \text{Book})$, transitivity fails. Since we do not have all such pairs, $S$ is not transitive. 7. **Explain reflexivity, symmetry, and non-transitivity:** - Reflexive: Each element in Set B relates to itself, so $(b,b) \in S$ for all $b \in B$. - Symmetric: For every $(b_1,b_2) \in S$, the pair $(b_2,b_1)$ is also in $S$. - Not transitive: There exist pairs $(b_1,b_2)$ and $(b_2,b_3)$ in $S$ but $(b_1,b_3)$ is not in $S$. **Final answers:** - Relation $R$ elements and inverse $R^{-1}$ as above. - $R$ is not a function. - Relation $S$ as above with explanations.