To solve the problem, we need to determine the set operation \((P - Q) \cap R\). Let's break this down step by step. ### Step 1: Understand the Sets We have three sets: - \(P = \{1, 2, u, v, w, x\}\) - \(Q = \{2, 3, u, v, w, 5, 6, y\}\) - \(R = \{2, 3, 4, v, x, y\}\) ### Step 2: Calculate \(P - Q\) The operation \(P - Q\) means we need to find the elements that are in set \(P\) but not in set \(Q\). 1. **List the elements of \(P\)**: \(1, 2, u, v, w, x\) 2. **Identify which of these elements are in \(Q\)**: - \(2\) is in \(Q\) - \(u\) is in \(Q\) - \(v\) is in \(Q\) - \(w\) is in \(Q\) - \(x\) is not in \(Q\) - \(1\) is not in \(Q\) 3. **Remove the elements of \(P\) that are in \(Q\)**: - From \(P\), we remove \(2, u, v, w\). - The remaining elements are \(1\) and \(x\). Thus, we have: \[ P - Q = \{1, x\} \] ### Step 3: Calculate \((P - Q) \cap R\) Now we need to find the intersection of the set \(P - Q\) with set \(R\). 1. **Identify the elements of \(P - Q\)**: \(\{1, x\}\) 2. **List the elements of \(R\)**: \(2, 3, 4, v, x, y\) 3. **Find common elements between \(\{1, x\}\) and \(R\)**: - \(1\) is not in \(R\) - \(x\) is in \(R\) Thus, the intersection is: \[ (P - Q) \cap R = \{x\} \] ### Conclusion The final answer is: **C. \{x\}** ### Explanation of Other Options - **A. \{1, x\}**: This option is incorrect because while \(x\) is indeed in the intersection, \(1\) is not in \(R\), so it cannot be part of the intersection. - **B. \{x, y\}**: This option is incorrect because \(y\) is not in \(P - Q\) at all, so it cannot be included in the intersection. - **D. ɸ (the empty set)**: This option is incorrect because we found that \(x\) is indeed in the intersection, so the intersection is not empty. ### Revision Summary - **Set Difference**: \(P - Q\) includes elements in \(P\) that are not in \(Q\). - **Intersection**: \((P - Q) \cap R\) includes elements common to both sets. - **Final Result**: The intersection yielded \{x\} as the only common element. - **Check Elements**: Always verify which elements belong to each set when performing set operations.