Freshman Maths 1#1 Propositional Logic Ethiopian University

Introduction

This tutorial provides a step-by-step guide on propositional logic as introduced in the Ethiopian University course. Propositional logic is a fundamental concept in mathematics and computer science, serving as a basis for reasoning and problem-solving. This guide will help you understand the basic principles and applications of propositional logic.

Step 1: Understanding Propositions

• A proposition is a declarative statement that is either true or false, but not both.
• Examples of propositions:
  • "The sky is blue." (True)
  • "2 + 2 = 5." (False)
• Non-examples include questions or commands, such as "What time is it?" or "Close the door."

Step 2: Logical Connectives

• Logical connectives are used to combine propositions. The main types include:
  • Conjunction (AND): True if both propositions are true. Symbol: ∧
    • Example: P ∧ Q
  • Disjunction (OR): True if at least one proposition is true. Symbol: ∨
    • Example: P ∨ Q
  • Negation (NOT): Inverts the truth value of a proposition. Symbol: ¬
    • Example: ¬P
  • Implication (IF...THEN): True unless a true proposition leads to a false one. Symbol: →
    • Example: P → Q
  • Biconditional (IF AND ONLY IF): True when both propositions have the same truth value. Symbol: ↔
    • Example: P ↔ Q

Step 3: Truth Tables

• Truth tables are used to determine the truth value of compound propositions.
• To create a truth table:
  • List all possible truth values for the propositions.
  • Use logical connectives to find the resulting truth values.

Example: For propositions P and Q, the truth table for P ∧ Q is:

| P | Q | P ∧ Q | |-------|-------|-------| | True | True | True | | True | False | False | | False | True | False | | False | False | False |

Step 4: Applications of Propositional Logic

• Propositional logic is used in various fields, including:
  • Computer programming for control flow (if statements).
  • Mathematics for constructing proofs.
  • Philosophy for analyzing arguments.

Step 5: Common Pitfalls

• Remember that propositional logic does not account for the nuances of natural language.
• Be cautious with implications; they are only false when a true proposition leads to a false conclusion.

Conclusion

Propositional logic is a vital area of study that lays the groundwork for advanced topics in mathematics and computer science. By understanding propositions, logical connectives, and truth tables, you can enhance your reasoning skills and apply these concepts in practical scenarios. For further study, consider exploring more complex forms of logic, such as predicate logic or modal logic.