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trí tuệ nhân tạo,cao hoàng trụ,dhbkhcm Artificial Intelligence (2014 2015) Tutorial 4 Propositional Logic Question 1 A proposition or statement is a sentence which is either true or false In the follo[.]

Artificial Intelligence (2014-2015)

Tutorial 4 Propositional Logic

Question 1

A proposition or statement is a sentence which is either true or false

In the following sentences, which one is proposition?

a The sun is shining

b The sum of two prime numbers is even

c 3+4=7

d It rained in Austin, TX, on October 30, 1999

e x+y > 10

f Is it raining?

g Come to class!

h n is a prime number

Question 2

For each of the following propositions, construct a truth table and state whether the proposition is valid or satisfiable (you can just write one truth table with many columns)

a P ∧ ¬P

b P ∨ ¬P

c (P ⇒ Q)⇒(P ⇒ Q)

d (P ∨ Q)⇒(P ∧ Q)

Question 3

Show that the following formulas are tautologies:

a ¬(A ∧ B) ⇔ ¬A∨¬B

b A⇒B ⇔ ¬B ⇒¬A

c ((A⇒B) ∧ (B ⇒A)) ⇔ (A ⇔ B)

d (A ∨ B) ∧ (¬B ∨ C)⇒(A ∨ C)

Question 4

Solve the following case: “If the criminal had an accomplice, then he came in a car The criminal had no accomplice and did not have the key, or he had the key and an

accomplice The criminal had the key Did the criminal come in a car or not?”

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Question 5.

Formalise the following and, by writing truth tables for the premises and conclusion, determine whether the arguments are valid

a Either John isn’t stupid and he is lazy, or he’s stupid John is stupid

Therefore, John isn’t lazy

b The butler and the cook are not both innocent Either the butler is lying or the cook

is innocent Therefore, the butler is either lying or guilty

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