[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.