Propositional Logic in CS103
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Propositional Logic in CS103

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Questions and Answers

What is represented by propositional variables in propositional logic?

  • Concepts or ideas
  • Certain numerical values
  • Logical connectives
  • Propositions that can be true or false (correct)

    • Which logical connective is true if both propositions are true?

  • Logical Implication
  • Logical NOT
  • Logical AND (correct)
  • Logical OR

    • What does the logical implication p → q signify?

  • If p is false, q must also be false.
  • If p is true, then q must also be true. (correct)
  • If both p and q are true, the implication holds.
  • If q is true, then p is also true.

    • In propositional logic, what is the outcome of the expression p ∨ q when both p and q are false?

    <p>False</p> Signup and view all the answers

    Which of the following represents logical negation?

    <p>¬p</p> Signup and view all the answers

    What would be the truth value of ¬p if p is true?

    <p>False</p> Signup and view all the answers

    Which of the following statements about p → q is incorrect?

    <p>It is independent of the truth value of p.</p> Signup and view all the answers

    What does the truth table for p ∧ q indicate when p is true and q is false?

    <p>False</p> Signup and view all the answers

    Which of the following statements is a proposition?

    <p>The cat is on the roof.</p> Signup and view all the answers

    Which of these logically describes a command?

    <p>It cannot be true or false.</p> Signup and view all the answers

    What is the main focus of propositional logic?

    <p>Basic logical connectives and truth tables.</p> Signup and view all the answers

    Which statement cannot be classified as a proposition?

    <p>Do you want to go to the park?</p> Signup and view all the answers

    What primarily differentiates a proposition from non-propositional statements?

    <p>Propositions can be assigned a truth value.</p> Signup and view all the answers

    Which of the following statements is an example of gibberish according to the classification of propositions?

    <p>I am the walrus, goo goo g'joob.</p> Signup and view all the answers

    What is the purpose of the upcoming first-order logic session?

    <p>To reason about properties of multiple objects.</p> Signup and view all the answers

    What is an example of a non-propositional statement?

    <p>Will it rain tomorrow?</p> Signup and view all the answers

    What is the only scenario in which the implication p → q is false?

    <p>p is true and q is false</p> Signup and view all the answers

    In a biconditional statement p ↔ q, when is it true?

    <p>When both p and q are true or both are false</p> Signup and view all the answers

    What does the symbol ⊤ represent in propositional logic?

    <p>A value that is always true</p> Signup and view all the answers

    Which connective indicates that p must not have the same truth value as q?

    <p>Biconditional (p ↔ q)</p> Signup and view all the answers

    Which statement correctly describes operator precedence in propositional logic?

    <p>Negation has the highest precedence</p> Signup and view all the answers

    How is the statement (¬x) → ((y ∨ z) → (x ∨ (y ∧ z))) parsed according to operator precedence?

    <p>((¬x) → (y ∨ z)) → (x ∨ (y ∧ z))</p> Signup and view all the answers

    Which of the following best describes a true statement in propositional logic?

    <p>A statement that is logically valid in every case</p> Signup and view all the answers

    What does the disjunction p ∨ q represent in propositional logic?

    <p>At least one of either p or q is true</p> Signup and view all the answers

    What does ¬(p → q) simplify to?

    <p>p ∧ ¬q</p> Signup and view all the answers

    Which of the following correctly expresses De Morgan's Laws?

    <p>¬(p ∨ q) ≡ ¬p ∧ ¬q</p> Signup and view all the answers

    What must be true for the implication p → q to be false?

    <p>p is true, q is false</p> Signup and view all the answers

    How can you prove that p → q is true using its contrapositive?

    <p>By proving ¬q → p</p> Signup and view all the answers

    If p → q is equivalent to ¬(p ∧ ¬q), which of the following describes this relationship?

    <p>Negation of an implication</p> Signup and view all the answers

    What is the correct transformation using De Morgan's Laws for the expression ¬(p ∨ q)?

    <p>¬p ∧ ¬q</p> Signup and view all the answers

    Which statement is equivalent to the theorem, 'If x + y = 16, then x ≥ 8 or y ≥ 8'?

    <p>If x &lt; 8 and y &lt; 8, then x + y ≠ 16</p> Signup and view all the answers

    Which of the following expressions represents the incorrect equivalence?

    <p>¬(p → q) ≡ p → ¬q</p> Signup and view all the answers

    What is the correct interpretation of 'p if q' in propositional logic?

    <p>q → p</p> Signup and view all the answers

    Which expression correctly represents the statement 'If there is a velociraptor outside my apartment, but it can't open windows, I am not going to be eaten by a velociraptor'?

    <p>a ∧ ¬b → ¬e</p> Signup and view all the answers

    How does 'p only when q' translate in propositional logic?

    <p>p → q</p> Signup and view all the answers

    What is the negation of 'p ∧ q' expressed in propositional logic?

    <p>¬p ∨ ¬q</p> Signup and view all the answers

    Which of the following is a logical equivalence statement?

    <p>¬(p → q) ≡ p ∧ ¬q</p> Signup and view all the answers

    What does the expression p ∧ (p → q) evaluate to when both p and q are true?

    <p>True</p> Signup and view all the answers

    Under which condition is the expression p ∧ q considered false?

    <p>When p is false or q is false</p> Signup and view all the answers

    Which of the following statements about logical equivalence is true?

    <p>φ and ψ are equivalent if they always yield the same truth values.</p> Signup and view all the answers

    Study Notes

    Announcements

    • Problem session tonight from 7:00 - 7:50 in room 380-380X.
    • Problem Set 3 Checkpoint due now.
    • Problem Set 2 Solutions where distributed at the end of class.
    • Stable office hours locations will be announced on the website soon.

    Propositions

    • Any statement that is, by itself, either true or false.
    • Examples of propositions: “Puppies are cuter than kittens.” “Kittens are cuter than puppies.” “Usain Bolt can outrun everyone in this room.” “CS103 is useful for cocktail parties.” “This is the last entry on this list.”
    • Things that are not propositions:
      • Commands: “You should have put a ring on it.”
      • Questions: “Are you a single lady?”
      • Jibberish: “I am the walrus, goo goo g'joob”

    Propositional Logic

    • Mathematical system for reasoning about propositions and their relationships.
    • Formally encodes how the influence of the truth of one proposition affects the truth of another.
    • Determines if combinations of propositions are always, sometimes, or never true.
    • Determines whether certain combinations of propositions logically imply other combinations.

    Variables and Connectives

    • Composed of propositional variables and logical connectives.
    • Propositional variables represent propositions (p, q, r, s, etc.).
    • Connectives connect variables and encode how propositions are related.

    Propositional Variables

    • Each propositional variable represents a single proposition.

    Logical Connectives

    • Negation (¬): “not p”. ¬p is true only if p is false.
    • Conjunction (∧): “p and q”. p ∧ q is true if both p and q are true.
    • Disjunction (∨): “p or q”. p ∨ q is true if at least one of p or q is true (inclusive OR).
    • Implication (→): “if p is true, q is true as well”. The only way for p → q to be false is if p is true and q is false.
    • Biconditional (↔): “p if and only if q”. Either both p and q are true, or neither are true.
    • True (⊤): A value that is always true.
    • False (⊥): A value that is always false.

    Truth Tables

    • Graphical representation of all possible truth values for a logical connective.
    • Used to verify logical equivalences.

    Translating into Propositional Logic

    • Be careful about nuances of the English language.
    • Use symbolic notation to avoid ambiguity.

    Logical Equivalence

    • Two propositional logic statements φ and ψ are logically equivalent if they always have the same truth values.
    • This is represented by φ ≡ ψ.
    • We can replace any occurrence of φ in a propositional logic formula with ψ.

    De Morgan's Laws

    • ¬(p ∧ q) ≡ ¬p ∨ ¬q
    • ¬(p ∨ q) ≡ ¬p ∧ ¬q

    Negating Implications

    • ¬(p → q) ≡ p ∧ ¬q

    Proof by Contrapositive

    • To prove p → q, show that ¬q → ¬p.

    Analyzing Proof Techniques

    • Proof by contrapositive: demonstrates the logical equivalence of p → q and ¬q → ¬p.

    Theorem, Logic, and Proof

    • Theorem: If x + y = 16, then either x ≥ 8 or y ≥ 8.
    • Proof by contrapositive:
      • Assumption: x < 8 and y < 8
      • Conclusion: x + y ≠ 16

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    Related Documents

    Stanford Mathematical Logic - First Lecture (PDF)

    Description

    Test your understanding of propositional logic and its role in reasoning. This quiz covers the basics of propositions, their relationships, and examples relevant to CS103. Be prepared to differentiate between true statements and other forms of communication.

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