Propositional Logic Basics

Propositional Logic

• Definition: Propositional Logic is a branch of mathematics that deals with statements that can be either true (T) or false (F).
• Propositions: Statements that can be either true or false, denoted by p, q, r, etc.
• Connectives: Symbols used to combine propositions, including:
  • NOT (negation): ¬p (not p)
  • AND (conjunction): p ∧ q (p and q)
  • OR (disjunction): p ∨ q (p or q)
  • IF-THEN (implication): p → q (if p, then q)
  • IF AND ONLY IF (equivalence): p ⇔ q (p if and only if q)

If-Then Statements (Implication)

• Definition: An if-then statement is a proposition of the form "if p, then q", denoted by p → q.
• Meaning: If p is true, then q must also be true. If p is false, the statement is true regardless of the value of q.
• Truth Table:

• Properties:
  • Transitivity: If p → q and q → r, then p → r.
  • Contrapositive: p → q is equivalent to ¬q → ¬p.
  • Modus Ponens: If p → q and p, then q.

Propositional Logic

• Definition: Deals with statements that can be either true (T) or false (F)
• Propositions: Statements that can be either true or false, denoted by p, q, r, etc

Connectives

• NOT (negation): ¬p (not p)
• AND (conjunction): p ∧ q (p and q)
• OR (disjunction): p ∨ q (p or q)
• IF-THEN (implication): p → q (if p, then q)
• IF AND ONLY IF (equivalence): p ⇔ q (p if and only if q)

If-Then Statements (Implication)

• Definition: A proposition of the form "if p, then q", denoted by p → q
• Meaning: If p is true, then q must also be true; if p is false, the statement is true regardless of the value of q
• Truth Table: shows the possible combinations of p and q values and their corresponding implications

Properties of If-Then Statements

• Transitivity: If p → q and q → r, then p → r
• Contrapositive: p → q is equivalent to ¬q → ¬p
• Modus Ponens: If p → q and p, then q