Introduction to Logic Quiz

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Questions and Answers

What is the primary focus of logic in relation to mathematics?

  • Performing complex mathematical proofs
  • Exploring historical mathematical theories
  • Understanding the significance of sound decision-making (correct)
  • The systematic study of numbers and calculations

Which of the following is a simple proposition?

  • He runs every day and eats healthy.
  • The sky is blue, and the grass is green.
  • Jonathan likes to play video games. (correct)
  • The sun rises in the east or sets in the west.

Which statement correctly describes a truth value?

  • The determination of a proposition being either true or false (correct)
  • The opinion based assessment of a statement
  • The representative numerical value of a proposition
  • The classification status of a proposition as subjective

What type of sentence is 'Go get the car' classified as?

<p>An imperative sentence (C)</p> Signup and view all the answers

Which statement is NOT a proposition?

<p>Is it raining today? (A)</p> Signup and view all the answers

In terms of propositions, what distinguishes a compound proposition from a simple proposition?

<p>A compound proposition contains multiple ideas. (D)</p> Signup and view all the answers

What is true about the statement '8 is a prime number'?

<p>It is a false proposition. (A)</p> Signup and view all the answers

Which of the following best defines a proposition?

<p>A declarative sentence that can be judged as true or false (B)</p> Signup and view all the answers

What does the symbol ~ represent in logical connectives?

<p>Negation (C)</p> Signup and view all the answers

Which statement corresponds to the proposition p Λ q where p = Today is Sunday and q = The shop is closed?

<p>Today is Sunday and the shop is closed. (C)</p> Signup and view all the answers

What is the word form of the logical connective with the symbol V?

<p>Or (C)</p> Signup and view all the answers

Which statement correctly represents the implication p → q where p = Today is Sunday and q = The shop is closed?

<p>If today is Sunday, then the shop is closed. (D)</p> Signup and view all the answers

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?

<p>Joy watched the concert of Ben&amp;Ben or she studies for the test. (A)</p> Signup and view all the answers

What is the correct symbolic form of the statement 'Joy passed the test if and only if she studied for the test'?

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

Which of the following statements correctly represents negation?

<p>Today is not Sunday. (D)</p> Signup and view all the answers

Which logical connective would be used to express the statement 'The shop is closed and it is Sunday'?

<p>Conjunction (A)</p> Signup and view all the answers

What does the truth table represent?

<p>Relationships between logical propositions (B)</p> Signup and view all the answers

Which of the following statements is a tautology?

<p>p → (q ∨ p) (D)</p> Signup and view all the answers

What is the truth value of the compound proposition p ∧ q if p is TRUE and q is FALSE?

<p>FALSE (C)</p> Signup and view all the answers

What does the symbol ~ represent in logical propositions?

<p>Negation (C)</p> Signup and view all the answers

In the expression (p ∨ q) ∧ p, what is the outcome if both p and q are FALSE?

<p>FALSE (A)</p> Signup and view all the answers

Which statement describes a contradiction?

<p>A statement false in all contexts (D)</p> Signup and view all the answers

If p is TRUE and q is FALSE, what is the truth value of p ∨ q?

<p>TRUE (D)</p> Signup and view all the answers

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?

<p>FALSE (A)</p> Signup and view all the answers

What is the truth value of the conjunction $p \land q$ when both $p$ and $q$ are false?

<p>False (A)</p> Signup and view all the answers

When will the disjunction $p \lor q$ be false?

<p>When both $p$ and $q$ are false (C)</p> Signup and view all the answers

What is the truth value of the biconditional statement $p \leftrightarrow q$ when $p$ is true and $q$ is false?

<p>False (C)</p> Signup and view all the answers

What is the truth value of the implication (p → q) when p is TRUE and q is FALSE?

<p>FALSE (D)</p> Signup and view all the answers

What is the truth value of the implication $p \rightarrow q$ when $p$ is false and $q$ is true?

<p>True (D)</p> Signup and view all the answers

How many different combinations of truth values are possible for three propositions p, q, and r?

<p>8 (A)</p> Signup and view all the answers

If $p$ is true and $q$ is true, what will the truth value of $p \land q$ be?

<p>True (A)</p> Signup and view all the answers

For the truth table of implication $p \rightarrow q$, what scenario results in a false truth value?

<p>When $p$ is true and $q$ is false (A)</p> Signup and view all the answers

Which of the following statements is TRUE about a biconditional statement?

<p>It is TRUE when both propositions share the same truth value. (A)</p> Signup and view all the answers

What is the result of the expression $\sim p \land q$ when $p$ is false and $q$ is true?

<p>True (D)</p> Signup and view all the answers

In terms of conjunction, when does it hold a true truth value?

<p>When both propositions are TRUE (A)</p> Signup and view all the answers

Which of the following statements about the biconditional statement is accurate?

<p>It is true when $p$ and $q$ are both true or both false. (A)</p> Signup and view all the answers

What is the truth value of (p ∨ q) ∧ r when p is FALSE, q is TRUE, and r is TRUE?

<p>TRUE (D)</p> Signup and view all the answers

Which statement correctly describes disjunction?

<p>It has a TRUE truth value when at least one proposition is TRUE. (B)</p> Signup and view all the answers

Which of the following is an incorrect statement regarding the truth values of conjunction?

<p>It can be TRUE if one proposition is TRUE and the other is FALSE. (B)</p> Signup and view all the answers

What is the truth value of the statement

<p>FALSE (C)</p> Signup and view all the answers

Flashcards

Proposition

A declarative sentence that is either true or false, but not both.

Truth Value

The value of a proposition, either True (T) or False (F).

Simple Proposition

A statement that expresses a single idea.

Compound Proposition

A statement that combines two or more ideas, often using logical connectives.

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Logical Connectives

Words or symbols that combine propositions to form compound statements.

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Declarative Sentence

A sentence that states a fact, opinion, or command; can be judged as true or false.

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Interrogative Sentence

A sentence that asks a question.

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Imperative Sentence

A sentence that gives a command or request.

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Tautology

A compound proposition that is always TRUE, no matter the truth values of its components.

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Contradiction

A compound proposition that is always FALSE, no matter the truth values of its components.

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Contingency

A proposition that can be either TRUE or FALSE depending on the truth values of its components.

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Negation (~)

The opposite truth value of a proposition.

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Conjunction (Λ)

A compound proposition that is TRUE only when BOTH propositions are TRUE.

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Disjunction (V)

A compound proposition that is TRUE when at least ONE proposition is TRUE, and TRUE if BOTH are TRUE.

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Negation

The negation of a proposition is its opposite. It uses the symbol '~' and is read as 'not'.

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Conjunction

The conjunction of two propositions is true only if both propositions are true. It uses the symbol 'Λ' and is read as 'and'.

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Disjunction

The disjunction of two propositions is true if at least one proposition is true. It uses the symbol 'V' and is read as 'or'.

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Implication

The implication of two propositions is true unless the first proposition is true and the second proposition is false. It uses the symbol '→' and is read as 'if...then'.

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Biconditional Statement

A biconditional statement is true only if both propositions have the same truth value. It uses the symbol '↔' and is read as 'if and only if'.

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Symbolic Form to Sentence Form

This converts propositions expressed in symbols into grammatically correct sentences using logical connectives.

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Pronouns in Logical Connectives

Pronouns can be used in sentences to avoid repetition and maintain clarity in compound propositions.

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Structure Flexibility

The structure of a sentence with logical connectives can be adjusted while preserving the truth value.

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Implication (→)

A compound statement that is only false if the first part is true and the second part is false. Otherwise, it's true.

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Biconditional Statement (↔)

A compound statement that is true only when both parts have the same truth value (both true or both false).

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What is a truth table?

A table that displays all possible truth values of propositions and their combinations with logical connectives.

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What does 'conjunction' mean?

'Conjunction' combines two propositions with 'AND' (∧). It's only TRUE when BOTH propositions are TRUE.

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How does 'disjunction' work?

'Disjunction' combines propositions with 'OR' (∨). It's TRUE if at least ONE proposition is TRUE, including if BOTH are TRUE.

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What is an 'implication'?

'Implication' (→) combines two propositions with 'IF...THEN'. It's only FALSE if the first is TRUE and the second is FALSE.

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What is a 'biconditional'?

A 'biconditional' (↔) combines propositions with 'IF AND ONLY IF'. It's TRUE only when BOTH propositions have the SAME truth value.

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What is the truth value of '∼p'?

'∼p' is the negation of 'p'. It has the OPPOSITE truth value of 'p'.

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What is the truth value of 'p∧q'?

The truth value of 'p∧q' (conjunction) is TRUE only when BOTH 'p' and 'q' are TRUE.

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What is the truth value of 'pVq'?

The truth value of 'pVq' (disjunction) is TRUE when at least ONE of 'p' or 'q' is TRUE, or BOTH are TRUE.

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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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