suppose p is the set of all students at west high school, with a the set of all students on the archery team and b the set of all students in the military cadets. identify in words the students described by region (z) of the following figure. choose the correct answer below. a. all students at west high who are in the military cadets but are not on the archery team b. all students at west high who are on the archery team but are not in the military cadets c. all students at west high who are on the archery team and are in the military cadets d. all students at west high who are not on the archery team and are not in the military cadets

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# Brief Explanations:
To determine the set for region $ z $, we analyze the Venn diagram:
- Set $ A $ is students on the archery team.
- Set $ B $ is students in military cadets.
- Region $ z $ is within set $ B $ (military cadets) but outside set $ A $ (archery team). So it represents students at West High who are in military cadets but not on the archery team. Wait, no—wait, re - examining: Wait, the options: Let's re - check. Wait, the region $ z $ is in circle $ B $ (military cadets) and outside circle $ A $ (archery team)? No, wait, the circles: $ A $ is archery, $ B $ is military. The region $ z $ is in $ B $ but not in $ A $? Wait, no, looking at the options: Option B says "All students at West High who are on the archery team but are not in the military cadets"—no, wait, maybe I mixed up. Wait, no, the region $ z $ is in circle $ B $ (military) and outside circle $ A $ (archery)? No, wait the diagram: The rectangle is $ P $ (all West High students). Circle $ A $ is archery, circle $ B $ is military. Region $ z $ is in $ B $ and outside $ A $? Wait, no, the labels: $ x $ is in $ A $ not $ B $, $ y $ is in both, $ z $ is in $ B $ not $ A $, $ w $ is in neither. Wait, but the options: Option B: "All students at West High who are on the archery team but are not in the military cadets"—that would be $ x $. Option A: "All students at West High who are in the military cadets but are not on the archery team"—that's $ z $. Wait, no, I think I made a mistake earlier. Wait, the correct region $ z $: If $ A $ is archery, $ B $ is military, then $ z $ is in $ B $ (military) and not in $ A $ (archery). So that's students in military but not archery. But wait, the options: Wait, the option B is "All students at West High who are on the archery team but are not in the military cadets" (that's $ x $), option A is "All students at West High who are in the military cadets but are not on the archery team" (that's $ z $)? Wait, no, the user's diagram: Let me re - read the problem. The problem says: $ P $ is all students at West High, $ A $ is all students on archery team, $ B $ is all students in military cadets. Region $ z $ is in $ B $ and outside $ A $? Wait, the diagram shows circle $ A $ (archery) and circle $ B $ (military). The regions: $ x $ is in $ A $ not $ B $, $ y $ is in both, $ z $ is in $ B $ not $ A $, $ w $ is in neither. So region $ z $ is students in $ B $ (military) and not in $ A $ (archery). So the correct option is B? Wait, no, option B is "All students at West High who are on the archery team but are not in the military cadets"—no, that's $ x $. Wait, I'm confused. Wait, the options:

A. All students at West High who are in the military cadets but are not on the archery team

B. All students at West High who are on the archery team but are not in the military cadets

C. All students at West High who are on the archery team and are in the military cadets

D. All students at West High who are not on the archery team and are not in the military cadets

Wait, region $ z $ is in circle $ B $ (military) and outside circle $ A $ (archery). So that's students in military (B) but not in archery (A). So that's option A? Wait, no, option A says "in military but not archery"—yes. Wait, but earlier I thought option B, but no. Wait, the diagram: circle $ A $ (archery), circle $ B $ (military). Region $ z $ is inside $ B $ and outside $ A $. So the students in $ z $ are in military (B) and not in archery (A). So the correct option is A? Wait, no, the option A is "All students at West High who are in the military cadets but are not on the archery team"—that's exactly region $ z $. Wait, but I think I messed up the initial analysis. So the correct answer is A? Wait, no, looking at the options again:

Wait, the user's diagram: The two circles, $ A $ (archery) and $ B $ (military). Region $ z $ is in $ B $ and outside $ A $. So the set for $ z $ is $ B \setminus A $ (military cadets minus archery team). So the students are in military cadets but not on archery team. So option A.

# Answer:
A. All students at West High who are in the military cadets but are not on the archery team

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