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Higher Study | Engineering Mathematics | Mathematical Logic

Which of the following statements are true ? (A) A sentence α entails another sentence β  if β is true in few worlds where α is true. (B) Forward chaining and backward chaining are very natural reasoning algorithms for knowledge bases in Horn form. (C) Sound inference algorithms derive all sentences that are entailed. (D) Propositional logic does not scale to environments of unbounded size. Choose the correct answer from the options given below:
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?
Consider the first-order logic sentence ? ≡ ∃?∃?∃?∀?∀?∀?∀? ?(?, ?, ?, ?, ?, ?, ?) where ?(?, ?, ?, ?, ?, ?, ?) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose ? has a model with a universe containing 7 elements.
What is logically equivalent to the following statements? "I pass only if you pass" (Note that fail is equivalent to not pass.)
What is the correct translation of the following statement into mathematical logic? “Some real numbers are rational”
What is the logical translation of the following statement? “None of my friends are perfect.”
Given that B(a) means “a is a bear” F(a) means “a is a fish” and E(a, b) means “a eats b” Then what is the best meaning of ∀ x [F(x) → ∀ y (E(y, x) → b(y))]
Consider the statement “Not all that glitters is gold” Predicate glitters(x) is true if x glitters and predicate gold(x) is true if x is gold. What which one of the following logical formulate represents the above statement?
Consider the statements: I. A ⇔  A ∨ ~ A II. A ⇔ A ∨ A III. A ∨ B ⇒ A IV. A ⇒ A ∨ B Which of the above refers to tautologies?
Consider the following logical inferences. I1: If it rains then the cricket match will not be played. The cricket match was played. Inference: There was no rain. I2: If it rains then the cricket match will not be played. It did not rain Inference: The cricket match was played. Which of the following is TRUE?
Which one of the following is NOT logically equivalent to ¬ ∃x (∀ y (α) ∧ ∀ z(β))?
Which set does not represent the sides of a right angled triangle?
Which of the following is not a tautology?
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 one of the following well-formed formulae in predicate calculus is NOT valid?
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