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Questions and Answers
What is the truth value of the conjunction p ⋀ q when both p and q are true?
What is the truth value of the conjunction p ⋀ q when both p and q are true?
In the context of disjunction, what is the truth value of p v q when both p and q are false?
In the context of disjunction, what is the truth value of p v q when both p and q are false?
Which of the following best describes the conjunction operator?
Which of the following best describes the conjunction operator?
What does the disjunction operator do under the inclusive or definition?
What does the disjunction operator do under the inclusive or definition?
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How is the truth value of p ⋀ q determined?
How is the truth value of p ⋀ q determined?
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What is the result of the exclusive or operation, p⨁q, when both p and q are true?
What is the result of the exclusive or operation, p⨁q, when both p and q are true?
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In the truth table for the exclusive or operation, what is the result of p⨁q when p is false and q is true?
In the truth table for the exclusive or operation, what is the result of p⨁q when p is false and q is true?
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Which statement about the truth values of p v q is correct when both propositions are true?
Which statement about the truth values of p v q is correct when both propositions are true?
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When is the conditional statement p → q considered false?
When is the conditional statement p → q considered false?
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What circumstance would make the disjunction p v q false?
What circumstance would make the disjunction p v q false?
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What is meant by the exclusive or in the context of disjunction?
What is meant by the exclusive or in the context of disjunction?
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If p is false, what can be said about the truth value of the conditional statement p → q?
If p is false, what can be said about the truth value of the conditional statement p → q?
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In the expression p → q, what are p and q specifically called?
In the expression p → q, what are p and q specifically called?
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What is the output of the conditional statement p → q when both p and q are false?
What is the output of the conditional statement p → q when both p and q are false?
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Which of the following scenarios would result in a true exclusive or operation, p⨁q?
Which of the following scenarios would result in a true exclusive or operation, p⨁q?
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Which statement about the truth value of p → q is incorrect?
Which statement about the truth value of p → q is incorrect?
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What does the bi-conditional statement involve?
What does the bi-conditional statement involve?
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Which of the following correctly represents the bi-conditional of two propositions 𝑝 and 𝑞?
Which of the following correctly represents the bi-conditional of two propositions 𝑝 and 𝑞?
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When will the statement 𝑝 ↔ 𝑞 be true?
When will the statement 𝑝 ↔ 𝑞 be true?
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What is the truth value of 𝑝 ↔ 𝑞 when both 𝑝 and 𝑞 are false?
What is the truth value of 𝑝 ↔ 𝑞 when both 𝑝 and 𝑞 are false?
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In the statement 'You can take the flight if and only if you buy a ticket', what does 'if and only if' indicate?
In the statement 'You can take the flight if and only if you buy a ticket', what does 'if and only if' indicate?
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What is the overall implication of the truth table for 𝑝 and 𝑞?
What is the overall implication of the truth table for 𝑝 and 𝑞?
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In the context of logical operators, what is the significance of operator precedence?
In the context of logical operators, what is the significance of operator precedence?
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What does a false bi-conditional statement indicate about the propositions 𝑝 and 𝑞?
What does a false bi-conditional statement indicate about the propositions 𝑝 and 𝑞?
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What characterizes a proposition in logic?
What characterizes a proposition in logic?
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Which statement is NOT a characteristic of Discrete Mathematics?
Which statement is NOT a characteristic of Discrete Mathematics?
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Which logical argument is exemplified by the statement, "For every positive integer n, the sum of the positive integers not exceeding n is $n(n+1)/2$"?
Which logical argument is exemplified by the statement, "For every positive integer n, the sum of the positive integers not exceeding n is $n(n+1)/2$"?
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What is a practical application of logic in computer science?
What is a practical application of logic in computer science?
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Which example represents a proposition?
Which example represents a proposition?
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In what way is graph theory relevant to daily life?
In what way is graph theory relevant to daily life?
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How often is a presidential election held in a leap year if leap years occur every 4 years and presidential elections every 6 years?
How often is a presidential election held in a leap year if leap years occur every 4 years and presidential elections every 6 years?
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What role does counting play in discrete mathematics?
What role does counting play in discrete mathematics?
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What is the precedence of the conjunction operator?
What is the precedence of the conjunction operator?
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In the expression $p → q ∧ ¬p$, what is the result when $p = T$ and $q = F$?
In the expression $p → q ∧ ¬p$, what is the result when $p = T$ and $q = F$?
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Which operator has the lowest precedence?
Which operator has the lowest precedence?
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What is the result of the expression $(p ∨ ¬q) → (p ∧ q)$ when $p = F$ and $q = F$?
What is the result of the expression $(p ∨ ¬q) → (p ∧ q)$ when $p = F$ and $q = F$?
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Which logical operator is represented by the symbol ↔?
Which logical operator is represented by the symbol ↔?
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What do the conventional letters represent in propositional variables?
What do the conventional letters represent in propositional variables?
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Which logical connective is used to negate a proposition?
Which logical connective is used to negate a proposition?
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What is the truth value of the proposition ¬p when p is true?
What is the truth value of the proposition ¬p when p is true?
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Which of the following represents a compound proposition?
Which of the following represents a compound proposition?
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Which logical operator combines two propositions and evaluates to true if both are true?
Which logical operator combines two propositions and evaluates to true if both are true?
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Who first developed propositional calculus?
Who first developed propositional calculus?
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What is the truth table entry for ¬p if p is false?
What is the truth table entry for ¬p if p is false?
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Which of the following operations combines multiple propositions without changing their truth values?
Which of the following operations combines multiple propositions without changing their truth values?
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Study Notes
Foundation: Logic and Proofs
- Course name: Foundation: Logic and Proofs
- Instructor: Lourielene Baldomero
- Institution: Cavite State University, College of Engineering and Information Technology
What is Discrete Mathematics?
- A study of countable or separable mathematical structures
- Covers various topics applicable to everyday life
- Examples:
- Logic (identifying fallacies in arguments)
- Number theory (frequency of presidential elections in leap years)
- Counting (number of outfits from available clothes)
- Probability (lottery chances)
- Recurrences (mortgage interest over time)
- Graph theory (fastest route between locations)
What is Logic?
- The study of laws of thought and correct reasoning
- Fundamental to mathematical and automated reasoning
- Example: Sum of integers not exceeding n = n(n+1)/2
Importance of Logic to Computer Science
- Used in designing computer circuits and programs
- Crucial for verifying program correctness
Propositions
- The basic building blocks of logic
- Declarative sentences
- Either true or false, but not both
- Examples:
- 1 + 1 = 2
- 2 + 2 = 3
- Lourielene is not the first name of an instructor in COSC 50A.
- Manila is the capital of the Philippines.
- Other examples:
- What time is it?
- Our topic today is logic.
- COSC 50A is an easy subject...
- Read the document carefully.
- x + 1 = 2
- x + y = z
Propositional Variables
- Variables representing propositions
- Conventional letters: p, q, r, s, ...
- Truth values:
- True (T)
- False (F)
Propositional Calculus (Propositional Logic)
- Developed by Aristotle over 2300 years ago
- Studies how statements interact with each other
Compound Propositions
- New propositions formed from existing ones using logical operators
- Combine one or more propositions to form complex statements
Logical Connectives
- Operators used to create new propositions from existing ones
- Examples:
- Negation
- Conjunction
- Disjunction
- Conditional
- Bi-conditional
Negation Operator
- Creates a new proposition from a single existing proposition
- Represented by -p (or ¬p)
- Means "It is not the case that p"
- Truth value is opposite of the original proposition
- Example:
- Truth table for -p:
- If p is T, then -p is F
- If p is F, then -p is T
- Truth table for -p:
Conjunction Operator
- Represented by ∧
- True if both propositions are true, otherwise false
- Example: Truth table for p∧q
- If p is T and q is T, then p∧q is T
- Otherwise, p∧q is F
Disjunction Operator
- Represented by ∨
- True if at least one operand is true; false if both are false
- Inclusive OR: True if either or both are true
- Exclusive OR: True only if exactly one is true
- Example: Truth table for p∨q
- If p is T, or q is T, or both are T, then p∨q is T
- Otherwise, p∨q is F
Conditional Statements
- Represented by →
- True unless a true hypothesis leads to a false conclusion
- "If p, then q"
- p = hypothesis, q = conclusion
- Example: Truth table for p→q
- If p is T and q is F, then p→q is F
- Otherwise, p→q is T
Bi-conditional Operator
- Represented by ↔
- True if both propositions have the same truth value; false otherwise
- "p if and only if q"
- Example: Truth table for p↔q
- If p and q are both T, or both F, then p↔q is T -Otherwise, p↔q is F
Precedence of Logical Operators
- Order of operations for evaluating complex propositions
- Negation (highest precedence)
- Conjunction
- Disjunction
- Implication
- Bi-conditional (lowest precedence)
Complex Compound Propositions
- Construct truth tables to analyze compound proposition involving logical operators
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Description
Test your understanding of discrete mathematics and logic concepts in this quiz focused on foundational principles. Explore the connections between logic, proof techniques, and their applications in computer science. Challenge yourself with practical examples and theoretical questions.
