Propositional Logic

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I keep going in circles trying to figure out the answer. Please help! 2. makes the most sense to me, but I am not sure what to do with the part with the -->

Which of the following are entailed by the sentence (A V B) Λ (¬C V ¬D V E)?

1. (A V B)
2. (A V B V C) Λ (B Λ C Λ D --> E)
3. (A V B) Λ (¬D V E)

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What is the precise definition of "entail" in this context?

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I keep going in circles trying to figure out the answer. Please help! 2. makes the most sense to me, but I am not sure what to do with the part with the -->

Which of the following are entailed by the sentence (A V B) Λ (¬C V ¬D V E)?

1. (A V B)
2. (A V B V C) Λ (B Λ C Λ D --> E)
3. (A V B) Λ (¬D V E)

Then think about what B Λ C Λ D --> E means and why that would result from the original sentence. It might help to think about this in terms of set theory and draw yourself a Venn diagram.

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Then think about what B Λ C Λ D --> E means and why that would result from the original sentence. It might help to think about this in terms of set theory and draw yourself a Venn diagram.

My reasoning was simple. #2 was the only choice with variables A, B, C, D, and E and there is nothing as far as I can see that would cancel out a variable.

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What is the precise definition of "entail" in this context?

"can be logically inferred"

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$KB \models \alpha$, where KB the Knowledge Base, and $\alpha$ is the statement that has a model with KB,

So, "(A V B) Λ (¬C V ¬D V E)" ~ KB = { (A V B), (¬C V ¬D V E) }

and since KB = TRUE, then (A V B) = TRUE, and (¬C V ¬D V E) = TRUE

1. TRUE, because (A V B) is TRUE in the KB

2. TRUE, because (A V B V C) = TRUE since (A V B) = TRUE in the KB, and (B Λ C Λ D --> E) = (¬B V ¬C V ¬D V E) = TRUE since (¬C V ¬D V E) = TRUE in the KB

3. Not sure, you have to revise Boolean Calculus, Wikipedia: Prepositional Calculus

Edited by khaled
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You might be over-analyzing... Check the definition of 'entailed'. (HINT: It doesn't mean a complete list of assertions)

EDIT: Ok, I'll add one thing-- If your question(assertion) is true, which of the 3 answers will definitely also be true?

Edited by Marqq
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You might be over-analyzing... Check the definition of 'entailed'. (HINT: It doesn't mean a complete list of assertions)

EDIT: Ok, I'll add one thing-- If your question(assertion) is true, which of the 3 answers will definitely also be true?

I kicked myself after thinking deeper about entailment. This whole time I've been trying to prove equivelancy between two equations when I really needed to just prove which one was always true when the first is true. I plugged all the equatinos into Excel and changed the values of A, B, C, D, and E and everytime the problem was true, #2 was also true. I did this a few days ago but got frustrated because I was thinking that when the first is false the others needed to be false also...which is not entailment.

Thanks for all the help!

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