Podcast
Questions and Answers
What is represented by propositional variables in propositional logic?
What is represented by propositional variables in propositional logic?
Which logical connective is true if both propositions are true?
Which logical connective is true if both propositions are true?
What does the logical implication p → q signify?
What does the logical implication p → q signify?
In propositional logic, what is the outcome of the expression p ∨ q when both p and q are false?
In propositional logic, what is the outcome of the expression p ∨ q when both p and q are false?
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Which of the following represents logical negation?
Which of the following represents logical negation?
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What would be the truth value of ¬p if p is true?
What would be the truth value of ¬p if p is true?
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Which of the following statements about p → q is incorrect?
Which of the following statements about p → q is incorrect?
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What does the truth table for p ∧ q indicate when p is true and q is false?
What does the truth table for p ∧ q indicate when p is true and q is false?
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Which of the following statements is a proposition?
Which of the following statements is a proposition?
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Which of these logically describes a command?
Which of these logically describes a command?
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What is the main focus of propositional logic?
What is the main focus of propositional logic?
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Which statement cannot be classified as a proposition?
Which statement cannot be classified as a proposition?
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What primarily differentiates a proposition from non-propositional statements?
What primarily differentiates a proposition from non-propositional statements?
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Which of the following statements is an example of gibberish according to the classification of propositions?
Which of the following statements is an example of gibberish according to the classification of propositions?
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What is the purpose of the upcoming first-order logic session?
What is the purpose of the upcoming first-order logic session?
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What is an example of a non-propositional statement?
What is an example of a non-propositional statement?
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What is the only scenario in which the implication p → q is false?
What is the only scenario in which the implication p → q is false?
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In a biconditional statement p ↔ q, when is it true?
In a biconditional statement p ↔ q, when is it true?
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What does the symbol ⊤ represent in propositional logic?
What does the symbol ⊤ represent in propositional logic?
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Which connective indicates that p must not have the same truth value as q?
Which connective indicates that p must not have the same truth value as q?
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Which statement correctly describes operator precedence in propositional logic?
Which statement correctly describes operator precedence in propositional logic?
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How is the statement (¬x) → ((y ∨ z) → (x ∨ (y ∧ z))) parsed according to operator precedence?
How is the statement (¬x) → ((y ∨ z) → (x ∨ (y ∧ z))) parsed according to operator precedence?
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Which of the following best describes a true statement in propositional logic?
Which of the following best describes a true statement in propositional logic?
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What does the disjunction p ∨ q represent in propositional logic?
What does the disjunction p ∨ q represent in propositional logic?
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What does ¬(p → q) simplify to?
What does ¬(p → q) simplify to?
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Which of the following correctly expresses De Morgan's Laws?
Which of the following correctly expresses De Morgan's Laws?
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What must be true for the implication p → q to be false?
What must be true for the implication p → q to be false?
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How can you prove that p → q is true using its contrapositive?
How can you prove that p → q is true using its contrapositive?
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If p → q is equivalent to ¬(p ∧ ¬q), which of the following describes this relationship?
If p → q is equivalent to ¬(p ∧ ¬q), which of the following describes this relationship?
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What is the correct transformation using De Morgan's Laws for the expression ¬(p ∨ q)?
What is the correct transformation using De Morgan's Laws for the expression ¬(p ∨ q)?
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Which statement is equivalent to the theorem, 'If x + y = 16, then x ≥ 8 or y ≥ 8'?
Which statement is equivalent to the theorem, 'If x + y = 16, then x ≥ 8 or y ≥ 8'?
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Which of the following expressions represents the incorrect equivalence?
Which of the following expressions represents the incorrect equivalence?
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What is the correct interpretation of 'p if q' in propositional logic?
What is the correct interpretation of 'p if q' in propositional logic?
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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'?
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'?
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How does 'p only when q' translate in propositional logic?
How does 'p only when q' translate in propositional logic?
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What is the negation of 'p ∧ q' expressed in propositional logic?
What is the negation of 'p ∧ q' expressed in propositional logic?
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Which of the following is a logical equivalence statement?
Which of the following is a logical equivalence statement?
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What does the expression p ∧ (p → q) evaluate to when both p and q are true?
What does the expression p ∧ (p → q) evaluate to when both p and q are true?
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Under which condition is the expression p ∧ q considered false?
Under which condition is the expression p ∧ q considered false?
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Which of the following statements about logical equivalence is true?
Which of the following statements about logical equivalence is true?
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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
Studying That Suits You
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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.
