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[M5-MAIN] Logic PowerPoint

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[M5-MAIN] Logic PowerPoint

Module Overview

  • Logic Technology Driven by Innovation

    • Institutions involved: FEU Alabang, FEU Diliman, FEU Tech

Propositions and Operations on Propositions

  • Propositions are foundational to logic and mathematics.

    • Involves technology-driven innovation in logical frameworks.

Mathematical Ideas

  • Mathematical Statements

    • Fundamental principles used in logic and reasoning.

Mathematical Logic

  • Definition: Derived from Greek "logos", meaning study or reason.

  • Nature of Logic:

    • Science and art of correct thinking.

    • Systematic body of logical truths governing reasoning.

Introduction to Logic

  • Formal Study: Systematic study of valid inference and reasoning.

    • Applied in philosophy, mathematics, semantics, and computer science.

  • Examines:

    • Forms of arguments.

    • Valid forms versus fallacies.

    • Purpose: Distinguishing good from bad arguments.

Logic Overview

  • Propositions:

    • Statements that are either true (T) or false (F).

    • Truth values correspond to binary values: 1 (true), 0 (false).

Defining Propositions

  • Characteristics:

    • Declarative sentences that declare facts.

    • Cannot be questions, instructions, or opinions.

  • Examples:

    • Toronto is the capital of Canada. (Yes)

    • Read this carefully. (No)

    • 1 + 2 = 3. (Yes)

    • x + 1 = 2. (No)

    • What time is it? (No)

Compound Statements

  • Formation:

    • Generated by combining or negating statements.

    • Component Statements: Individual statements forming a compound statement.

  • Connectives: "and", "or", "not", "if...then".

Statement Classification Exercise

  • Identify statements, compound statements, or neither:

    • Diagram connecting components (if...then)

    • Simple statement examples.

Negations

  • Definition: Refusal or denial of a statement.

  • Examples:

    • Statement: The number 9 is odd.

    • Negation: The number 9 is not odd.

    • True/False relationship in negation.

Inequality Symbols and Negations

  • Symbolism:

    • Less than, greater than, less than or equal to, greater than or equal to relations.

  • Negation Examples for inequalities.

Simplifying Logic with Symbols

  • Propositional Variables: Represent propositions (p, q, r).

  • Truth Values: T, F.

  • Connectives and their types:

    • Conjunction (AND, ∧)

    • Disjunction (OR, ∨)

    • Negation (NOT, ~ ).

Logical Operators (Connectives)

  • Types of Logical Connectives:

    • Negation (NOT) ~

    • Conjunction (AND) ∧

    • Disjunction (OR) ∨

    • Exclusive or (XOR)

    • Implication (if...then) →

    • Bi-conditional (if and only if).

Implications in Logic

  • Implication: p → q can be read as:

    • "If p then q"

    • Various equivalent statements.

Translating Symbols to Words

  • Practice in rewriting logical statements from symbols to verbal expressions.

Quantifiers in Statements

  • Universal Quantifiers: All, each, no, none.

  • Existential Quantifiers: Some, there exists.

Negation of Quantified Statements

  • Patterns of negation in quantified statements.

Truth Values Overview

  • Importance in logic.

  • Examples showing true and false statements.

Truth Tables and Equivalent Statements

  • How component statements' truth values determine compound statements.

  • Example of a conjunction truth table.

Conditional Statements

  • Characteristics of Conditionals:

    • Compound statements using "if...then".

    • Establishing truth table for conditional statements.

Converse, Inverse, and Contrapositive

  • Relationship between conditional statements and their converses/inverses.

  • Examples demonstrating equivalence.

Rewording Conditional Statements

  • Exercises on transforming conditional statements into "if...then" form.

Negation of Conditional Statements

  • Expression of the negation of conditional statements.

References

  • Various educational materials and publications supporting the course.

  • Authors: Barcelona A.B., Oronce O.A., Verzosa D.M.B. et al.