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What is the primary focus of logic in relation to mathematics?
What is the primary focus of logic in relation to mathematics?
Which of the following is a simple proposition?
Which of the following is a simple proposition?
Which statement correctly describes a truth value?
Which statement correctly describes a truth value?
What type of sentence is 'Go get the car' classified as?
What type of sentence is 'Go get the car' classified as?
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Which statement is NOT a proposition?
Which statement is NOT a proposition?
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In terms of propositions, what distinguishes a compound proposition from a simple proposition?
In terms of propositions, what distinguishes a compound proposition from a simple proposition?
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What is true about the statement '8 is a prime number'?
What is true about the statement '8 is a prime number'?
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Which of the following best defines a proposition?
Which of the following best defines a proposition?
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What does the symbol ~ represent in logical connectives?
What does the symbol ~ represent in logical connectives?
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Which statement corresponds to the proposition p Λ q where p = Today is Sunday and q = The shop is closed?
Which statement corresponds to the proposition p Λ q where p = Today is Sunday and q = The shop is closed?
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What is the word form of the logical connective with the symbol V?
What is the word form of the logical connective with the symbol V?
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Which statement correctly represents the implication p → q where p = Today is Sunday and q = The shop is closed?
Which statement correctly represents the implication p → q where p = Today is Sunday and q = The shop is closed?
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In the sentence form of the statement p V q, what does it mean when p = Joy watched the concert of Ben&Ben and q = Joy studies for the test?
In the sentence form of the statement p V q, what does it mean when p = Joy watched the concert of Ben&Ben and q = Joy studies for the test?
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What is the correct symbolic form of the statement 'Joy passed the test if and only if she studied for the test'?
What is the correct symbolic form of the statement 'Joy passed the test if and only if she studied for the test'?
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Which of the following statements correctly represents negation?
Which of the following statements correctly represents negation?
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Which logical connective would be used to express the statement 'The shop is closed and it is Sunday'?
Which logical connective would be used to express the statement 'The shop is closed and it is Sunday'?
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What does the truth table represent?
What does the truth table represent?
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Which of the following statements is a tautology?
Which of the following statements is a tautology?
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What is the truth value of the compound proposition p ∧ q if p is TRUE and q is FALSE?
What is the truth value of the compound proposition p ∧ q if p is TRUE and q is FALSE?
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What does the symbol ~ represent in logical propositions?
What does the symbol ~ represent in logical propositions?
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In the expression (p ∨ q) ∧ p, what is the outcome if both p and q are FALSE?
In the expression (p ∨ q) ∧ p, what is the outcome if both p and q are FALSE?
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Which statement describes a contradiction?
Which statement describes a contradiction?
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If p is TRUE and q is FALSE, what is the truth value of p ∨ q?
If p is TRUE and q is FALSE, what is the truth value of p ∨ q?
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What is the result of the expression (~p ∧ r) → (q ∧ s) if p is TRUE, r is FALSE, q is TRUE, and s is TRUE?
What is the result of the expression (~p ∧ r) → (q ∧ s) if p is TRUE, r is FALSE, q is TRUE, and s is TRUE?
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What is the truth value of the conjunction $p \land q$ when both $p$ and $q$ are false?
What is the truth value of the conjunction $p \land q$ when both $p$ and $q$ are false?
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When will the disjunction $p \lor q$ be false?
When will the disjunction $p \lor q$ be false?
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What is the truth value of the biconditional statement $p \leftrightarrow q$ when $p$ is true and $q$ is false?
What is the truth value of the biconditional statement $p \leftrightarrow q$ when $p$ is true and $q$ is false?
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What is the truth value of the implication (p → q) when p is TRUE and q is FALSE?
What is the truth value of the implication (p → q) when p is TRUE and q is FALSE?
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What is the truth value of the implication $p \rightarrow q$ when $p$ is false and $q$ is true?
What is the truth value of the implication $p \rightarrow q$ when $p$ is false and $q$ is true?
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How many different combinations of truth values are possible for three propositions p, q, and r?
How many different combinations of truth values are possible for three propositions p, q, and r?
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If $p$ is true and $q$ is true, what will the truth value of $p \land q$ be?
If $p$ is true and $q$ is true, what will the truth value of $p \land q$ be?
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For the truth table of implication $p \rightarrow q$, what scenario results in a false truth value?
For the truth table of implication $p \rightarrow q$, what scenario results in a false truth value?
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Which of the following statements is TRUE about a biconditional statement?
Which of the following statements is TRUE about a biconditional statement?
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What is the result of the expression $\sim p \land q$ when $p$ is false and $q$ is true?
What is the result of the expression $\sim p \land q$ when $p$ is false and $q$ is true?
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In terms of conjunction, when does it hold a true truth value?
In terms of conjunction, when does it hold a true truth value?
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Which of the following statements about the biconditional statement is accurate?
Which of the following statements about the biconditional statement is accurate?
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What is the truth value of (p ∨ q) ∧ r when p is FALSE, q is TRUE, and r is TRUE?
What is the truth value of (p ∨ q) ∧ r when p is FALSE, q is TRUE, and r is TRUE?
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Which statement correctly describes disjunction?
Which statement correctly describes disjunction?
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Which of the following is an incorrect statement regarding the truth values of conjunction?
Which of the following is an incorrect statement regarding the truth values of conjunction?
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What is the truth value of the statement
What is the truth value of the statement
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Study Notes
Introduction to Logic
- Logic is the study of reasoning and principles governing valid arguments and sound decision-making.
- It's a foundational tool in mathematics, philosophy, computer science, and everyday problem-solving.
Objectives
- Define logic and its significance in mathematics and reasoning.
- Identify different types of statements and their logical properties.
- Apply logical connectives to construct compound statements.
- Analyze arguments for validity using truth tables and logical equivalences.
- Solve problems involving logical reasoning.
Key Concepts
- Proposition: A declarative sentence that is either true or false, but not both. Represented by letters (e.g., p, q, r, s).
- Truth Value: The value of a proposition, either True (T) or False (F).
Examples of Propositions and Their Truth Values
- Manila is the capital of the Philippines: Proposition, TRUE.
- Dogs are mammals: Proposition, TRUE.
- It is Tuesday?: NOT a proposition (interrogative).
- Go get the car.: NOT a proposition (imperative).
- 8 is a prime number.: Proposition, FALSE.
- This is a nice car.: NOT a proposition (subjective).
Types of Propositions
-
Simple Proposition: A statement conveying a single idea.
- Example: Jonathan likes to play video games.
-
Compound Proposition: A statement conveying two or more ideas.
- Example: Jonathan likes to play video games and always stays up late.
Logical Connectives
-
Negation (~): "not"
- Example: ~p: "Today is not Sunday."
-
Conjunction (Λ): "and"
- Example: p ^ q: "Today is Sunday and the shop is closed."
-
Disjunction (V): "or"
- Example: p V q: "Today is Sunday or the shop is closed."
-
Implication (→): "if...then"
- Example: p → q: "If today is Sunday, then the shop is closed."
-
Biconditional (↔): "if and only if"
- Example: p ↔ q: "Today is Sunday if and only if the shop is closed."
Summary of Logical Connectives
| Connectives | Word Form | Statement | Symbolic Form |
|---|---|---|---|
| Negation | not | not p | ~p |
| Conjunction | and | p and q | p ^ q |
| Disjunction | or | p or q | p V q |
| Implication | if...then | if p...then q | p → q |
| Biconditional | if and only if | p if and only if q | p ↔ q |
Truth Tables
- A diagram used to organize, present, and show the relationships between the truth values of propositions involving logical connectives.
Tautology, Contradiction, Contingency
- Tautology: A compound proposition that generates a TRUE value in all possible combinations of simple propositions.
- Contradiction: A compound proposition that generates a FALSE value in all possible combinations of simple propositions.
- Contingency: A proposition that is neither a tautology nor a contradiction.
Truth Values of Logical Propositions
- Negation: The opposite truth value of the proposition.
- Conjunction (e.g p ^ q): TRUE only if both p and q are TRUE. Otherwise, FALSE.
- Disjunction (e.g p V q): TRUE if either p,q or both are TRUE; FALSE if both are FALSE.
- Implication (p → q): FALSE only if p is TRUE and q is FALSE; otherwise TRUE.
- Biconditional (p ↔ q): TRUE only if p and q share the same truth value. Otherwise, FALSE.
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Description
Test your understanding of the fundamental principles of logic. This quiz covers key concepts, types of statements, and the application of logical connectives. Assess your ability to analyze arguments and determine their validity using truth values and truth tables.
