Answer to Question #95271 in Discrete Mathematics for Aqilah Hannani

Question

The famous detective TVTHREE, Kara Singh Walla was called in to solve a baffling
murder mystery. He determined the following facts:

A. Alan, the murdered man, was killed by a blow on the head with brass
candlestick.

B. Either Alan’s wife or a maid, Sara, was in the dining room at the time of the
murder.

C. If the cook was in the kitchen at the time of the murder, then the butler killed
Alan with fatal dose of strychnine.

D. If the Alan’s wife was in the dining room at the time of the murder, then the
chauffeur killed Alan.

E. If the cook was not in the kitchen at the time of the murder, then Sara was not
in the dining room when the murder committed.

F. If Sara was in the dining room at the time the murder was committed, then the
wine steward killed Alan.

a. Write each primary statement above using the variables p, q, r, s, t, and u.
b. Write all the hypotheses or premises based on the statements given.

Answer 1

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.