Advanced Truth Tables Explained

Nov 8, 2024

Truth Tables Tutorial Part 3

Introduction

  • Series on truth tables, continuing from videos made in 2016.
  • Aim: Addressing unanswered questions from previous videos.
    • Part 2: Introduced different notations and operators like biconditional and exclusive OR.
    • Part 3: Advanced concepts with three propositions (P, Q, R).

Expert Level Truth Tables

Three Propositions (P, Q, R)

  • Standard truth table filled out by instructor, but sometimes students may need to list possibilities.
  • Standard Order for List:
    • True, True, True
    • True, True, False
    • True, False, True
    • True, False, False
    • False, True, True
    • False, True, False
    • False, False, True
    • False, False, False
  • Duplication of Order: Copy twice for cases where third proposition changes from true to false.

Combining Propositions

  • Example: P or Q
    • Check P and Q columns for at least one T.
  • Example: P or Q or R
    • Similar approach using P or Q column as a reference.

Order of Operations

  • P or Q or R without parentheses
    • Treated as equivalent to (P or Q) or R due to left to right operation.
  • Multiplication analogy with addition for OR operation
  • AND takes precedence over OR, similar to multiplication over addition in arithmetic.

Example with AND and OR

  • P or Q and R
    • Without parentheses, implies (P or (Q and R)).
    • Demonstrates importance of operations order.

Using IF, THEN

  • If Q and R, then P or Q
    • Requires intermediate columns for clarity.
    • If Q and R column T exists, compare P or Q columns.

Key Concepts

Order of Operations (Logical)

  1. NOT comes first.
  2. AND (multiplication) comes before OR (addition).
  3. IF THEN follows.
  4. IF and ONLY IF comes last.

Additional Points

  • Multiple order scenarios could give different answers.
  • Tautology: A statement always true.

Conclusion

  • Practice individual operations before composition.
  • Teachers should provide parentheses to avoid confusion.
  • Use implied parentheses for operations if not provided.
    • Follow the logical order: NOT, AND, OR, IF THEN, IF and ONLY IF.