Introduction to Truth Tables and Conditional Statements

Introduction to Logic and Truth Tables

• Basic Concepts
  • Logic: The study of reasoning and argumentation in mathematics.
  • Truth Tables: Tools used to determine the truth values of logical expressions based on the truth values of their components.
  • Statements: Any declarative sentence that can be classified as true (T) or false (F).
  • Variables: Commonly denoted as p, q, etc., to represent statements in logical expressions.

Truth Values

• Each statement can take on a truth value:
  • True (T)
  • False (F)
• Negations change the truth value:
  • Rule for Negations:
  • If p is True (T), then the negation (¬p) is False (F)
  • If p is False (F), then the negation (¬p) is True (T)

Conditional Statements

• A conditional statement has the form "if p, then q", denoted as p → q.
• Truth Conditions for Conditional Statements:
  • True only when
  • both p and q are True
  • p is False (regardless of q's value)
  • False only when
  • p is True and q is False
  • Example Truth Values:
  • p: T, q: T ⇒ p → q: T
  • p: T, q: F ⇒ p → q: F
  • p: F, q: T ⇒ p → q: T
  • p: F, q: F ⇒ p → q: T

Completing a Truth Table

• To complete a truth table for a given logical expression, identify the variables and apply the rules for negations and conditional statements.
• Example: For expressions involving p and q, identify all combinations of truth values:

|  p  |  q  |  ¬p  |  p ∧ q  |  p → q  |  ¬p → q  |
|-----|-----|------|--------|---------|--------|
|  T  |  T  |  F   |   T    |    T    |   T    |
|  T  |  F  |  F   |   F    |    F    |   T    |
|  F  |  T  |  T   |   F    |    T    |   T    |
|  F  |  F  |  T   |   F    |    T    |   F    |

• Interpreting Rows:
  • The truth value of each expression under its respective columns helps to derive the overall truth of the logical statement.
• Note that additional columns can be added to clarify intermediate steps.

Final Thoughts

• Remember that understanding the structure and rules of each logic statement is crucial for accurate computations in truth tables.
• Recognizing how the truth value changes based on logical operators will aid in math problems involving logic.