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## Here's the Solution to this Question

a).. We let the fact A as it is in order to can use all 6 variables on the rest of the facts. Also, fact A is not related to any of the other facts.

P: Alan's wife was in the dining room at the time of the murder.

Q: Sara the maid was in the dining room at the time of the murder.

R: The cook was in the kitchen at the time of the murder.

S: The butler killed Alan with fatal dose of strychnine.

T: The chauffeur killed Alan.

U: The wine steward killed Alan.

Then we can rewrite the provided facts as below:

B). P ∨ Q

C). R → S

D). P → T

E). ¬ R → ¬ Q

F). Q → U

b). First of all, E) is equivalent with Q→ R being the contrapositive of the original statement.

This translates to: If Sara the maid was in the dining room at the time of the murder, then the cook was in the kitchen at the time of the murder.

Then, combining E) and C), according to hypothetical syllogism (transitivity): Q→S.

This translates to: If Sara the maid was in the dining room at the time of the murder, then the butler killed Alan with

fatal dose of strychnine.

Then, since D) is equivalent with ¬ P ∨ T, we can combine it with B) and, according to the resolution rule: T ∨ Q

This translates to: The chauffeur killed Alan or Sara the maid was in the dining room at the time of the murder.

We can also combine B) with E) (which is equivalent with ¬ Q ∨ R) according to resolution: P ∨ R

This translates to: Alan's wife was in the dining room at the time of the murder or the cook was in the kitchen at the time of the murder.

We can also combine B) with F) (which is equivalent with ¬ Q ∨ U) according to resolution to obtain: P ∨ U

This translates to: Alan's wife was in the dining room at the time of the murder or the wine steward killed Alan.

We can also combine C) and the equivalent form of E) according to resolution: ¬Q ∨ S

This translates to: Sara the maid was not in the dining room at the time of the murder or the butler killed Alan with fatal dose of strychnine.