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Mathematical Logic and Set Theory

Mathematical Logic and Set Theory Quiz

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following is a tautology in propositional logic?

  1. ¬(P ∨ Q)
  2. (P → Q) → ¬P
  3. P ∨ (¬P ∧ Q)
  4. ¬(P ∧ Q) → (¬P ∨ ¬Q)
Question 2 Multiple Choice (Single Answer)

What is the negation of the statement "∀x ∈ R, x^2 ≥ 0"?

  1. ∃x ∈ R, x^2 < 0
  2. ∀x ∈ R, x^2 > 0
  3. ∃x ∈ R, x^2 ≤ 0
  4. ∀x ∈ R, x^2 ≠ 0
Question 3 Multiple Choice (Single Answer)

Which of the following is a valid inference rule in propositional logic?

  1. Modus Ponens
  2. Modus Tollens
  3. Hypothetical Syllogism
  4. Disjunctive Syllogism
Question 4 Multiple Choice (Single Answer)

What is the power set of the set {1, 2, 3}?

  1. {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, ∅}
  2. {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
  3. {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}
  4. {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, ∅, {∅}}
Question 5 Multiple Choice (Single Answer)

Which of the following is an example of a bijection?

  1. The function f: R → R defined by f(x) = x^2
  2. The function g: R → R defined by g(x) = x + 1
  3. The function h: R → R defined by h(x) = sin(x)
  4. The function j: R → R defined by j(x) = |x|
Question 6 Multiple Choice (Single Answer)

What is the cardinality of the set of all subsets of the set {1, 2, 3}?

  1. 2^3
  2. 3^2
  3. 2^4
  4. 3^3
Question 7 Multiple Choice (Single Answer)

Which of the following is an example of a countable set?

  1. The set of all natural numbers
  2. The set of all rational numbers
  3. The set of all real numbers
  4. The set of all functions from R to R
Question 8 Multiple Choice (Single Answer)

What is the complement of the set {x ∈ R | x^2 < 4}?

  1. {x ∈ R | x^2 ≥ 4}
  2. {x ∈ R | x^2 > 4}
  3. {x ∈ R | x^2 ≤ 4}
  4. {x ∈ R | x^2 ≠ 4}
Question 9 Multiple Choice (Single Answer)

Which of the following is an example of a valid argument in first-order logic?

  1. All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
  2. All dogs are mammals. All mammals are animals. Therefore, all dogs are animals.
  3. Some birds can fly. Tweety is a bird. Therefore, Tweety can fly.
  4. No cats are dogs. Garfield is a cat. Therefore, Garfield is not a dog.
Question 10 Multiple Choice (Single Answer)

What is the negation of the statement "∃x ∈ R, x^2 = 2"?

  1. ∀x ∈ R, x^2 ≠ 2
  2. ∃x ∈ R, x^2 > 2
  3. ∀x ∈ R, x^2 < 2
  4. ∃x ∈ R, x^2 ≤ 2
Question 11 Multiple Choice (Single Answer)

Which of the following is an example of a deductive argument?

  1. The sky is blue. Therefore, the grass is green.
  2. All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
  3. I saw a black cat yesterday. Therefore, all cats are black.
  4. I like chocolate ice cream. Therefore, everyone likes chocolate ice cream.
Question 12 Multiple Choice (Single Answer)

What is the cardinality of the set of all functions from the set {1, 2, 3} to the set {a, b, c}?

  1. 3^3
  2. 3^2
  3. 2^3
  4. 2^2
Question 13 Multiple Choice (Single Answer)

Which of the following is an example of a non-constructive proof?

  1. Proof by contradiction
  2. Proof by mathematical induction
  3. Proof by exhaustion
  4. Proof by construction
Question 14 Multiple Choice (Single Answer)

What is the intersection of the sets {1, 2, 3} and {2, 3, 4}?

  1. {1, 2, 3, 4}
  2. {1, 2, 3}
  3. {2, 3}
  4. {1, 4}
Question 15 Multiple Choice (Single Answer)

Which of the following is an example of a countably infinite set?

  1. The set of all natural numbers
  2. The set of all rational numbers
  3. The set of all real numbers
  4. The set of all functions from R to R
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