Question 1003091: Given: If Alex does not score well on his final exams and Alex does not earn an academic scholarship, the he does not go to college. If Alex does not study, then he does not score well on his final exams. if Alex gets involved in too many extracurricular activities, then Alex does not study. Alex gets involved in too many extracurricular activities. Alex goes to college.
Let C represent: "Alex goes to college"
Let E represent: "Alex earns an academic scholarship"
Let F represent: "Alex scores well on his final exams"
Let S represent: "Alex studies"
Let X represent: "Alex gets involved in too many extracurricular activities"
Prove: Alex earns an academic scholarship
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
| Original Statement | Translation |
|---|
| If Alex does not score well on his final exams and Alex does not earn an academic scholarship, the he does not go to college. | (~F * ~E) -> ~C | | If Alex does not study, then he does not score well on his final exams. | ~S -> ~F | | if Alex gets involved in too many extracurricular activities, then Alex does not study. | X -> ~S | | Alex gets involved in too many extracurricular activities. | X | | Alex goes to college. | C |
We have these premises
(~F * ~E) -> ~C
~S -> ~F
X -> ~S
X
C
The goal is to prove E. One way to do it is to use a derivation table and follow the rules of inference/rules of replacement
| Number | Statement | Lines Used | Reason |
|---|
| 1 | (~F * ~E) -> ~C | | | | 2 | ~S -> ~F | | | | 3 | X -> ~S | | | | 4 | X | | | | 5 | C | | | | :. | E | | | | 6 | ~S | 3,4 | MP | | 7 | ~F | 2,6 | MP | | 8 | ~~C | 5 | DN | | 9 | ~(~F * ~E) | 1,8 | MT | | 10 | ~~F v ~~E | 9 | DM | | 11 | F v E | 10 | DN | | 12 | E | 11,7 | DS |
Abbreviations used:
MP = Modus Ponens
MT = Modus Tollens
DN = Double Negation
DM = De Morgan's Law
DS = Disjunctive Syllogism
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