Answer to Question #308959 in Discrete Mathematics for Zain

Question

1)Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.

a) ∀x(C(x) → F(x))

b) ∀x(C(x) ∧ F(x))

c) ∃x(C(x) → F(x))

2) Somie, a leader of the underworld, was killed by one of his own band of four henchmen. Detective Sharp interviewed the men and determined that all were lying except for one. He deduced who killed Somie on the basis of the following statements:

a. Socko: Lefty killed Somie.

b. Fats: Muscles didn’t kill Somie.

c. Lefty: Muscles was shooting craps with Socko when Somie was knocked off.

d. Muscles: Lefty didn’t kill Somie.

Who did kill Somie?

Answer 1

1 a. For every x, if x is a comedian then x is funny.

b. For all x, x is a comedian and x is funny.

c. There exist an x, such that if x is a comedian then x is funny.

2 Let S be Socko killed Somie

F be Fats killed Somie

L be Lefty Killed Somie

M be Muscle Killed Somie

Scenario 1-Lets assume Socko said the truth and everyone else lied

a. L

b. M

c. ~M, ~S

d. L

Thus inconsistent because of part b and c (Muscle killed Somie and Muscle did not .

Scenario 2. Lets assume Fats said the truth

a. ~L

b. M

c. M, S

d. L

Not consistent because of part a and d (Lefty did not killed Somie and Lefty killed Somie)

Scenario 3. Lets assume Lefty said the truth

a. ~L

b. M

c. ~M, ~S

d. L

Not consistent because of a. and d and b. and c. all have contradicting statements.

Scenario 4. Lets assume Muscle said the truth

a. ~L

b. M

c. M, S

d. ~L

Thus the only consistent one is Scenario 4, when Muscle is saying the truth thus Muscle killed Somie.