KTU Notes: Logic Fundamentals

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Module 1: Fundamentals of Logic

• Proposition (Statement): A declarative sentence that is either true or false, but not both.
• Example: "Margaret wrote 'Gone with the Wind'" is a proposition.
• Combinatorics: A required course for sophomores.
• Truth Value: A proposition has a truth value of either true (1) or false (0).
• Statements that are not propositions do not have truth values.

Statements

• Examples of statements:
  • In 2003, George W. Bush was the president of the United States.
  • Fifteen is an even number.
  • If Jennifer is late for the party, then Zachary will be quite angry.
• Examples of non-statements:
  • x + 3 is a positive integer
  • What time is it?
  • As of June 30, 2003, Christine Marie Evert had won the French Open a record seven times

Basic Connectives and Truth Tables

• Negation (¬): The negation of a proposition p, denoted by ¬p, is the statement "p is not true". The truth value of ¬p is the opposite of the truth value of p.
• Truth Table (for negation):
  • If p is true, ¬p is false.
  • If p is false, ¬p is true.
• Conjunction (∧): The conjunction of p and q, denoted by p ∧ q, is true only if both p and q are true; otherwise, it is false.
• Disjunction (∨): The disjunction of p and q, denoted by p ∨ q, is false only if both p and q are false; otherwise it is true.

Compound Statements

• Implication (→): If p then q, denoted by p → q. This statement is false only when p is true and q is false.
• Biconditional (↔): If and only if, denoted by p ↔ q. This statement is true only when both p and q have the same truth value.

Tautology and Contradiction

• Tautology: A compound statement is always true.
• Contradiction: A compound statement is always false.

Logical Equivalences

• There are many logical equivalences involving conditional and biconditional statements.