Let's solve the problem **step by step** as requested. --- ### **Step 1: List all equivalence relations on \( A \).** **Explanation:** An equivalence relation on a set corresponds to a partition of that set. For \( A = \{a, b, c\} \), the partitions (and thus the equivalence relations) are: 1. **All elements in their own class:** - Partition: \( \{\{a\}, \{b\}, \{c\}\} \) - Relation: \( \{(a, a), (b, b), (c, c)\} \) 2. **One pair together, one separate:** - Partition: \( \{\{a, b\}, \{c\}\} \) - Relation: \( \{(a, a), (b, b), (c, c), (a, b), (b, a)\} \) - Partition: \( \{\{a, c\}, \{b\}\} \) - Relation: \( \{(a, a), (b, b), (c, c), (a, c), (c, a)\} \) - Partition: \( \{\{b, c\}, \{a\}\} \) - Relation: \( \{(a, a), (b, b), (c, c), (b, c), (c, b)\} \) 3. **All together:** - Partition: \( \{\{a, b, c\}\} \) - Relation: \( \{(a, a), (b, b), (c, c), (a, b), (b, a), (a, c), (c, a), (b, c), (c, b)\} \) **Summary:** There are **5 equivalence relations** on \( A \). --- ### **Step 2: For each relation provide the partition classes.** **Explanation:** The partition classes are given above for each relation. Let's write them clearly: 1. \( \{\{a\}, \{b\}, \{c\}\} \) 2. \( \{\{a, b\}, \{c\}\} \) 3. \( \{\{a, c\}, \{b\}\} \) 4. \( \{\{b, c\}, \{a\}\} \) 5. \( \{\{a, b, c\}\} \) --- ### **Final Answer:** #### **(a) List of all equivalence relations on \( A \):** - \( \{(a, a), (b, b), (c, c)\} \) - \( \{(a, a), (b, b), (c, c), (a, b), (b, a)\} \) - \( \{(a, a), (b, b), (c, c), (a, c), (c, a)\} \) - \( \{(a, a), (b, b), (c, c), (b, c), (c, b)\} \) - \( \{(a, a), (b, b), (c, c), (a, b), (b, a), (a, c), (c, a), (b, c), (c, b)\} \) #### **(b) Partition classes for each relation:** - \( \{\{a\}, \{b\}, \{c\}\} \) - \( \{\{a, b\}, \{c\}\} \) - \( \{\{a, c\}, \{b\}\} \) - \( \{\{b, c\}, \{a\}\} \) - \( \{\{a, b, c\}\} \) #### **(c) Are any of these partial order relations?** **Explanation:** A partial order must be **reflexive, antisymmetric, and transitive**. Equivalence relations are reflexive, symmetric, and transitive (not antisymmetric unless each class is a singleton). - The only candidate is the relation where each element is only related to itself: \( \{(a, a), (b, b), (c, c)\} \). This is the **equality relation**, which **is** a partial order (and a total order). **Final Answer:** **Yes, the relation \( \{(a, a), (b, b), (c, c)\} \) is a partial order (the equality relation).**

Let's approach this systematically: --- ### **Step 1: List all equivalence relations on \( A = \{a, b, c\} \).** **Explanation:** Equivalence relations on a set correspond to partitions of that set. For a set with 3 elements, the possible partitions are well-known: 1. **Each element in its own singleton:** - Partition: \(\{\{a\}, \{b\}, \{c\}\}\) - Relation: all pairs \((x, x)\) (reflexivity), plus no off-diagonal pairs. - Relation: \(\{(a, a), (b, b), (c, c)\}\) 2. **One pair together, the other separate:** - Partitions: a) \(\{\{a, b\}, \{c\}\}\) b) \(\{\{a, c\}, \{b\}\}\) c) \(\{\{b, c\}, \{a\}\}\) - Relations include all pairs within each block, plus the diagonal: - For \(\{\{a, b\}, \{c\}\}\): \(\{(a, a), (b, b), (c, c), (a, b), (b, a)\}\) - Similarly for others. 3. **All elements together:** - Partition: \(\{\{a, b, c\}\}\) - Relation: all pairs \((x, y)\) where \(x, y \in \{a, b, c\}\). **Total equivalence relations:** **5** --- ### **Step 2: Provide the partition classes for each relation.** **Answer:** 1. \(\{\{a\}, \{b\}, \{c\}\}\) 2. \(\{\{a, b\}, \{c\}\}\) 3. \(\{\{a, c\}, \{b\}\}\) 4. \(\{\{b, c\}, \{a\}\}\) 5. \(\{\{a, b, c\}\}\) --- ### **Are any of these relations partial orders?** **Explanation:** A partial order must be **reflexive, antisymmetric, and transitive**. - Equivalence relations are **reflexive, symmetric, and transitive**, but **not necessarily antisymmetric** unless each class is a singleton. - The relation where each element is only related to itself: \(\{(a, a), (b, b), (c, c)\}\), is **reflexive, antisymmetric, transitive** — i.e., a partial order (specifically, the equality relation). **Final conclusion:** Yes, **the relation with only diagonal pairs** is a partial order. --- ### **Final answer:** - **All equivalence relations** are the five listed above, corresponding to the partitions: 1. \(\{\{a\}, \{b\}, \{c\}\}\) 2. \(\{\{a, b\}, \{c\}\}\) 3. \(\{\{a, c\}, \{b\}\}\) 4. \(\{\{b, c\}, \{a\}\}\) 5. \(\{\{a, b, c\}\}\) - **Partial order relation among these:** The only partial order relation is the identity (equality) relation: \(\{(a, a), (b, b), (c, c)\}\). --- Let me know if you'd like further clarification!