Show that proposition C is a logical consequence of the formula

     A ∧ (A →(B ∨ C)) ∧ (B → ~A)  

Using truth tables.

(a) Let * be a Boolean operation defined as
If C = A * B then evaluate and fill in the blanks:
(i) A * A = _______
(ii) C * A = _______
(b) Solve the following boolean equations for the values of A, B and C:

Obtain the principal (canonical) conjunctive normal form of the propositional formula

  (p ∧ q) V (¬q ∧ r) 

Where ‘∧’ is logical and, ‘v’ is inclusive or and ¬ is negation.

Which one of the following predicate formulae is NOT logically valid?
Note that W is a predicate formula without any free occurrence of x.

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

Consider the first order predicate formula φ:

∀x[(∀z z|x ⇒ ((z = x) ∨ (z = 1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒ ((w = z) ∨ (z = 1)))]

Here 'a|b' denotes that 'a divides b', where a and b are integers. Consider the following sets:

S1.  {1, 2, 3, ..., 100}
S2.  Set of all positive integers
S3. Set of all integers

Which of the above sets satisfy φ?

Which of the following propositions is a tautology?

What is the converse of the following assertion?

  I stay only if you go. 

(a) Show that the formula [(~p ∨ q) ⇒ (q ⇒ p)] is not a tautology.
(b) Let A be a tautology and B be any other formula. Prove that (A ∨ B) is a tautology.

Let a, b, c, d be propositions. Assume that the equivalence a ↔ (b ∨ -b) and b ↔ c hold. Then the truth-value of the formula (a ∧ b) → (a ∧ c) ∨ d is always

Consider two well-formed formulas in prepositional logic

   F1: P ⇒ ¬P          F2: (P⇒¬P)∨(¬P⇒P)  

Which of the following statements is correct?

“If X then Y unless Z” is represented by which of the following formulas in prepositional logic? (“¬“, is negation, “∧” is conjunction, and “→” is implication)

(a) Uses Modus ponens (A, A →|= B) or resolution to show that the following set is inconsistent:

 (1) Q(x) → P(x)V ~ R(a)
 (2) R(a) ~ Q(a)
 (3) Q(a)
 (4) ~ P(y) 

Where x and y are universally quantified variables, a is a constant and P, Q, R are monadic predicates.

(b) Let S be the set of all integers and let n > 1 be a fixed integer. Define for a, b ∈ S, a R biff a-b is a multiple of n. Show that R is an equivalence relation and finds its equivalence classes for n = 5.

Consider the following first order logic formula in which R is a binary relation symbol.

 ∀x∀y (R(x, y)  => R(y, x)) 

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

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.

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

Which of the following well-formed formulas are equivalent?

Choose the correct alternatives (More than one may be correct). Indicate which of the following well-formed formulae are valid:

The following resolution rule is used in logic programming.

Derive clause (P ∨ Q) from clauses (P ∨ R), (Q ∨ ¬R) 

Which of the following statements related to this rule is FALSE?

Consider the following formula a and its two interpretations I₁ and I₂

α: (∀x)[Px ⇔ (∀y)[Qxy ⇔ ¬Qyy]] ⇒ (∀x)[¬Px]
I1: Domain: the set of natural numbers
    Px ≡ 'x is a prime number'
    Qxy ≡ 'y divides x'
I2: same as I1 except that Px = 'x is a composite number'.

Which of the following statements is true?

Which of the following is a valid first order formula? (Here α and β are first order formulae with x as their only free variable)

The following propositional statement is (P → (Q v R)) → ((P v Q) → R)

Identify the correct translation into logical notation of the following assertion.

      "Some boys in the class are taller than all the girls"  

Note: taller(x,y) is true if x is taller than y.

Which one of the following well formed formulae is a tautology?

Which one of the following well-formed formulae in predicate calculus is NOT valid?

Let Graph(x) be a predicate which denotes that x is a graph. Let Connected(x) be a predicate which denotes that x is connected. Which of the following first order logic sentences DOES NOT represent the statement: “Not every graph is connected”?

P and Q are two propositions. Which of the following logical expressions are equivalent?

I. P∨∼Q
II. ∼(∼P∧Q)
III. (P∧Q)∨(P∧∼Q)∨(∼P∧∼Q)
IV. (P∧Q)∨(P∧∼Q)∨(∼P∧Q)

Let fsa and pda be two predicates such that fsa(x) means x is a finite state automaton, and pda(y) means that y is a pushdown automaton. Let equivalent be another predicate such that equivalent (a, b) means a and b are equivalent. Which of the following first order logic statements represents the following:

Each finite state automaton has an equivalent pushdown automaton

Suppose the predicate F(x, y, t) is used to represent the statement that person x can fool person y at time t. Which one of the statements below expresses best the meaning of the formula ∀x∃y∃t(¬F(x,y,t))?

Which one of the following options is CORRECT given three positive integers x,y and z, and a predicate

What is the correct translation of the following statement into mathematical logic?

“Some real numbers are rational”