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13 Jan 2026, 10:35
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At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?
(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.
(2) Every student enrolled in Chemistry is also enrolled in Biology.
(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.
(2) Every student enrolled in Chemistry is also enrolled in Biology.
13 Jan 2026, 10:36
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Official Solution:
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?
(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.
This means anyone in Chemistry must also be in Biology because if a student were in Chemistry but not in Biology, then that student would be a counterexample to the statement. Since such a student cannot exist, all students in Chemistry must also be in Biology. So every Chemistry student is also in Biology. Since Chemistry has more than 25 students, at least those 25+ students are definitely in both courses. Sufficient.
(2) Every student enrolled in Chemistry is also enrolled in Biology.
This says the same thing: anyone in Chemistry must also be in Biology because if someone belonged to Chemistry without belonging to Biology, the condition would be violated. Therefore every Chemistry student must already be a Biology student. So again, every Chemistry student is in both courses. Since Chemistry has more than 25 students, there are definitely students in both. Sufficient.
Answer: D
At a certain university, each of two advanced courses, Biology and Chemistry, has more than 25 students enrolled. Is there at least one student enrolled in both courses?
(1) Every student who is not enrolled in Biology is also not enrolled in Chemistry.
This means anyone in Chemistry must also be in Biology because if a student were in Chemistry but not in Biology, then that student would be a counterexample to the statement. Since such a student cannot exist, all students in Chemistry must also be in Biology. So every Chemistry student is also in Biology. Since Chemistry has more than 25 students, at least those 25+ students are definitely in both courses. Sufficient.
(2) Every student enrolled in Chemistry is also enrolled in Biology.
This says the same thing: anyone in Chemistry must also be in Biology because if someone belonged to Chemistry without belonging to Biology, the condition would be violated. Therefore every Chemistry student must already be a Biology student. So again, every Chemistry student is in both courses. Since Chemistry has more than 25 students, there are definitely students in both. Sufficient.
Answer: D
12 Feb 2026, 04:51
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Bunuel - Does it depend on the assumption that there are no students who are not present in both Bio & Chemistry?
Bunuel
