A famous detective was called in to solve a baffling murder mystery. He determined there was only one cause of death along with the following facts:
a.Mr. Harold, the murdered man, was killed by a blow on the head with a brass candlestick.
b.Either his wife, Mrs. Harold, 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 Harold with a fatal dose of strychnine.
d.If his wife was in the dining room at the time of the murder, then the chauffeur killed Mr. Harold.
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 was committed.
f.If Sara was in the dining room at the time the murder was committed, then the wine steward killed Mr. Harold.
Convert the statements to symbols making sure you define each letter that you use. If it is possible to deduce the identity of the murderer from the above facts, then do so and explain your reasoning. If not, explain why.
A: candlestick killed
B: wife or Sara in DR
C: cook in kitchen ==> butler use strychnine
D: wife in DR ==> chauffeur killed
E: Sara in DR ==> cook in kitchen
F: Sara in DR ==> steward killed
~C (strychnine not used), so cook not in kitchen
so, ~E: Sara not in DR (so, also, ~F)
so, B: wife in DR
so, D: chauffeur
and A: with candlestick
Let's assign symbols to each statement in order to analyze the murder mystery more systematically:
a. Mr. Harold was killed by a blow on the head with a brass candlestick.
- Let H be the statement "Mr. Harold was killed by a blow on the head with a brass candlestick."
b. Either his wife, Mrs. Harold, or a maid, Sara, was in the dining room at the time of the murder.
- Let W be the statement "Mrs. Harold was in the dining room at the time of the murder."
- Let S be the statement "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 Harold with a fatal dose of strychnine.
- Let C be the statement "The cook was in the kitchen at the time of the murder."
- Let B be the statement "The butler killed Harold with a fatal dose of strychnine."
d. If his wife was in the dining room at the time of the murder, then the chauffeur killed Mr. Harold.
- Let C be the statement "The chauffeur killed Mr. Harold."
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 was committed.
- Since this statement is a conditional, we can assign symbols to its two parts:
- Let ¬C be the statement "The cook was not in the kitchen at the time of the murder."
- Let ¬S be the statement "Sara was not in the dining room when the murder was committed."
f. If Sara was in the dining room at the time of the murder, then the wine steward killed Mr. Harold.
- Let W be the statement "The wine steward killed Mr. Harold."
Based on these symbols, we now have the following statements:
a. H
b. W ∨ S
c. (C → B)
d. (W → C)
e. (¬C → ¬S)
f. (S → W)
To determine the identity of the murderer, we can use deductive reasoning based on the given facts. By analyzing the statements, we can identify contradictions and inconsistencies.
From statements b and e, we can infer:
¬W → ¬S
Using this information, we can consider the case where both W and S are false:
¬W ∧ ¬S
From statements c and d, we can infer:
¬W → ¬C
Using this information, we can consider the case where both W and C are false:
¬W ∧ ¬C
From statement e, we can infer:
¬C → S
Using this information, we can consider the case where both C and S are true:
C ∧ S
Now, let's combine the cases to identify the possible combinations:
Case 1: ¬W ∧ ¬S ∧ ¬C
Case 2: ¬W ∧ ¬C ∧ S
Case 3: ¬W ∧ C ∧ S
In all three cases, Mr. Harold was killed by a blow on the head with a brass candlestick (H). However, none of the statements definitively point to a single murderer. Therefore, we cannot deduce the identity of the murderer from the given facts.