Propositional and First Order Logic.

Question 41

Which one of these first-order logic formula is valid?

• ∀x(P(x) => Q(x)) => (∀xP(x) => ∀xQ(x))
• ∃x(P(x) ∨ Q(x)) => (∃xP(x) => ∃xQ(x))
• ∃x(P(x) ∧ Q(x)) <=> (∃xP(x) ∧ ∃xQ(x))
• ∀x∃y P(x, y) => ∃y∀x P(x, y)

Question 42

Which of the following first order formula is logically valid? Here α(x) is a first order formula with x as a free variable, and β is a first order formula with no free variable.

• [β→(∃x,α(x))]→[∀x,β→α(x)]
• [∃x,β→α(x)]→[β→(∀x,α(x))]
• [(∃x,α(x))→β]→[∀x,α(x)→β]
• [(∀x,α(x))→β]→[∀x,α(x)→β]

Question 43

Which of the following is the negation of [∀ x, α → (∃y, β → (∀ u, ∃v, y))]

• [∃ x, α → (∀y, β → (∃u, ∀ v, y))]
• [∃ x, α → (∀y, β → (∃u, ∀ v, ¬y))]
• [∀ x, ¬α → (∃y, ¬β → (∀u, ∃ v, ¬y))]
• [∃ x, α ʌ (∀y, β ʌ (∃u, ∀ v, ¬y))]

Question 44

What is the converse of the following assertion?

I stay only if you go.

• I stay if you go

• If I stay then you go

• If you do not go then I do not stay

• If I do not stay then you go

Question 45

Which of the following propositions is a tautology?

• (p ∨ q) → p
• p ∨ (q → p)
• p ∨ (p → q)
• p → (p → q)

Question 46

Which of the following is false? Read ∧ as AND, ∨ as OR, ∼ as NOT, → as one way implication and ↔ as two way implication.

• ((x→y) ∧ x)→ y
• ((∼x→y) ∧ (∼x→∼y))→ x
• (x→ (x ∨ y))
• ((x ∨ y) ↔ (∼x→∼y))

Question 47

Let apple(x) be the predicate that x is an apple. Let green(x) be the predicate that the color of x is green. Which of the following statements does not represent the given statement? "Not every apple is green"

• A
• B
• C
• D

Question 48

How many given logical equivalence(s) is/are false?

(I) [Tex]{\\displaystyle \\neg (p\\iff q)\\equiv p\\iff \\neg q}[/Tex]
(II) [Tex]{\\displaystyle (p\\implies q)\\wedge (p\\implies r)\\equiv p\\implies (q\\vee r)} [/Tex]
(III) [Tex]{\\displaystyle (p\\implies q)\\vee (p\\implies r)\\equiv p\\implies (q\\wedge r)}  [/Tex]
(IV) [Tex] {\\displaystyle (p\\implies r)\\wedge (q\\implies r)\\equiv (p\\vee q)\\implies r} [/Tex]
(V) [Tex]{\\displaystyle (p\\implies r)\\vee (q\\implies r)\\equiv (p\\vee q)\\implies r}  [/Tex]

• 0
• 2
• 3
• None of these

Question 49

What is the correct translation of the following statement into mathematical logic? “Every student who walks talks”

(I) ∀x ((student(x) & walk (x)) → talk (x)))
(II) ∀x (student(x) → (walk (x) → talk (x)))
(III)  ¬ ∃x ((student(x) & walk (x)) & ¬(talk (x))))

• Only (I)
• Only (II)
• Only (II) and (III)
• All (I), (II), and (III)

Question 50

Which of the following compound proposition is not tautology?

• ((p→q)∧(q →r))→(p→r)
• ((p∧q)∧(q∧r))→(p∧r)
• ((p⊕q)∧(q⊕r))→(p⊕r)
• (P V Q) Λ (P V R) → P V (Q Λ R)