Consider a vocabulary with only four propositions A, B, C and D. How many…
2018
Consider a vocabulary with only four propositions A, B, C and D. How many models are there for the following sentence?
\(¬𝐴∨¬𝐵∨¬𝐶∨¬𝐷\)
- A.
7
- B.
8
- C.
15
- D.
16
Attempted by 226 students.
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Correct answer: C
Count the models for the sentence ¬A ∨ ¬B ∨ ¬C ∨ ¬D.
Total number of truth assignments for four propositions: 2^4 = 16.
When is the disjunction ¬A ∨ ¬B ∨ ¬C ∨ ¬D false?
The disjunction is false exactly when every disjunct is false, i.e., when A, B, C and D are all true. There is exactly one such assignment.
Therefore, number of models = total assignments − assignments that falsify the sentence = 16 − 1 = 15.
Total assignments: 16
Assignments that make the formula false: 1 (A, B, C, D all true)
Models (satisfying assignments): 15
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