1. **Problem:** Determine if the relation $R = \{(A,B) \in P \times P : A = B\}$ on the power set $P$ of $S = \{2,3,7\}$ is reflexive, symmetric, and/or transitive. 2. **Recall definitions:** - Reflexive: For all $A$, $(A,A) \in R$. - Symmetric: If $(A,B) \in R$, then $(B,A) \in R$. - Transitive: If $(A,B) \in R$ and $(B,C) \in R$, then $(A,C) \in R$. 3. **Check reflexivity:** Since $R$ contains all pairs where $A = B$, for every subset $A$ of $S$, $(A,A) \in R$. So $R$ is reflexive. 4. **Check symmetry:** If $(A,B) \in R$, then $A = B$. Hence $(B,A)$ also satisfies $B = A$, so $(B,A) \in R$. So $R$ is symmetric. 5. **Check transitivity:** If $(A,B) \in R$ and $(B,C) \in R$, then $A = B$ and $B = C$, so $A = C$. Hence $(A,C) \in R$. So $R$ is transitive. **Final answer:** The relation $R$ is reflexive, symmetric, and transitive.