Discrete Structures I
Due: June 1, 11:59PM Atlantic Standard Time
Please write down your name and B00 number on your solution
Problem 1 (5pts): For this exercise, it is easier not to use truth tables.
a) Suppose that A → ((B ∧ C) → D) is false. What are the truth values of A, B, C, and D?
b) Suppose that (A → B) ∨ ((C ∧ (D ∧ E)) → F ) is false. What are the truth values of A, B, C, D, E,
and F ?
Problem 2 (5pts): Prove the equivalences below without using truth tables. Instead, use the rules of
logical equivalence given in class. Justify your reasoning.
a) ∼ (P ∨ (Q ∨ R)) ≡∼ P ∧ (∼ Q∧ ∼ R).
b) P → (Q → S) ≡ (P ∧ Q) → S.
c) ((P → Q) → P ) → P ≡ 1.
Problem 3 (6pts): Reduce the following statements to the fewest number of literals.
a) (∼ P ∧ ∼ Q∧ ∼ R) ∨ (∼ P ∧ Q ∧ R) ∨ (∼ P ∧ Q∧ ∼ R) ∨ (P ∧ ∼ Q∧ ∼ R) ∨ (P ∧ ∼ Q ∧ R)
b) (P ∧ ∼ Q∧ ∼ R) ∨ (∼ P ∧ ∼ R∧ ∼ S) ∨ (∼ P ∧ ∼ R ∧ S) ∨ (∼ P ∧ R∧ ∼ S) ∨ (∼ P ∧ Q ∧ R ∧ S) ∨ (P ∧ ∼
Q ∧ R∧ ∼ S))
Problem 4 (3pts): Formalize the following argument and show that it is valid using the inference rules
If Jose took the jewelery or Mrs. Krasov lied, then a crime was committed. Mrs. Krasov was
not in town. If a crime was committed, then Mrs. Krasov was in town. Therefore Jose did not
take the jewelery.
Problem 5 (4pts):
quantifiers ∀ and ∃.
Formalize the following statements using the connectives ∧, ∨, ∼, and → and the
a) Every non-negative integer strictly less than 4 is an integer squared.
b) For every real numbers x and y there is a rational number q such that if x is strictly smaller than y
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