Difference between revisions of "1968 AHSME Problems/Problem 10"
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<math>\text{(A) Some students are fraternity members.} \quad\\ | <math>\text{(A) Some students are fraternity members.} \quad\\ | ||
| − | \text{(B) Some fraternity | + | \text{(B) Some fraternity members are not students.} \quad\\ |
\text{(C) Some students are not fraternity members.} \quad\\ | \text{(C) Some students are not fraternity members.} \quad\\ | ||
\text{(D) No fraternity member is a student.} \quad\\ | \text{(D) No fraternity member is a student.} \quad\\ | ||
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== Solution == | == Solution == | ||
| − | <math>\fbox{C}</math> | + | If some students are dishonest, we know that they must not be fraternity members, because if they were members, then they would be honest. This conclusion aligns with answer choice <math>\fbox{C}</math>. |
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| + | Choice (A) is incorrect, because if there were no fraternity members, proposition II would be [https://en.wikipedia.org/wiki/Vacuous_truth vacuously true], and dishonest students are still allowed to exist within the rules of the problem, satisfying propostion I. | ||
| + | |||
| + | Choice (B) is incorrect, because dishonest students can still exist (proposition I) while some other students (the fraternity members, maybe some others) are honest (proposition II). | ||
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| + | Choice (D) is incorrect by the same reasoning used against choice (B). | ||
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| + | Choice (E) is incorrect, because it is the [[contrapositive]] of choice (D), so the two are logically equivalent. (Choice (D) could be restated as "if someone is a fraternity member, that someone is not a student," and choice (E) could be restated as "if someone is a student, that someone is not a fraternity member.") | ||
== See also == | == See also == | ||
| − | {{AHSME box|year=1968|num-b=9|num-a=11}} | + | {{AHSME 35p box|year=1968|num-b=9|num-a=11}} |
[[Category: Introductory Logic Problems]] | [[Category: Introductory Logic Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 18:59, 17 July 2024
Problem
Assume that, for a certain school, it is true that
I: Some students are not honest. II: All fraternity members are honest.
A necessary conclusion is:
Solution
If some students are dishonest, we know that they must not be fraternity members, because if they were members, then they would be honest. This conclusion aligns with answer choice 
.
Choice (A) is incorrect, because if there were no fraternity members, proposition II would be vacuously true, and dishonest students are still allowed to exist within the rules of the problem, satisfying propostion I.
Choice (B) is incorrect, because dishonest students can still exist (proposition I) while some other students (the fraternity members, maybe some others) are honest (proposition II).
Choice (D) is incorrect by the same reasoning used against choice (B).
Choice (E) is incorrect, because it is the contrapositive of choice (D), so the two are logically equivalent. (Choice (D) could be restated as "if someone is a fraternity member, that someone is not a student," and choice (E) could be restated as "if someone is a student, that someone is not a fraternity member.")
See also
| 1968 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
