"Introduction to truth tables with negations, conjunctions, or disjunctions"
Introduction to Logic and Truth Tables
- Definition of a Statement: A statement is a sentence that is either true (T) or false (F).
- Example:
- p: "The chicken is in the barn"
- q: "The farmer is asleep"
- Truth Value: The truth value of a statement indicates whether the statement is true or false.
- Example: If "The chicken is in the barn" (p) is false, then the truth value of p is F.
Truth Tables
- Purpose: A truth table shows how the truth values of statements determine the truth values of combined statements.
- Basic Symbols:
- p, q: Individual statements
- ∼ : Negation (not)
- ∧ : Conjunction (and)
- ∨ : Disjunction (or)
Types of Statements
Negation (∼q)
- Reads as "not q"
- Truth value:
- If q is true, ∼q is false.
- If q is false, ∼q is true.
- Interpretation: "When q is false, ∼q is true."
Conjunction (p ∧ q)
- Reads as "p and q"
- Definition: True if both p and q are true; otherwise, false.
- Important Note: The order of p and q does not affect the outcome, meaning p ∧ q is the same as q ∧ p.
- Interpretation: "When p is true and q is false, p ∧ q is false."
Disjunction (p ∨ q)
- Reads as "p or q"
- Definition: True if either p or q is true; false only when both are false.
- Important Note: The order of p and q does not affect the outcome, meaning p ∨ q is the same as q ∨ p.
- Interpretation: "When p is true and q is false, p ∨ q is true."
Example of a Truth Table with Values
| p | q | ∼q | p ∨ q | p ∧ q |
|---|---|---|---|---|
| T | T | F | T | T |
| T | F | T | T | F |
| F | T | F | T | F |
| F | F | T | F | F |
- This table illustrates how various combinations of truth values of p and q affect the outcomes of ∼q, p ∨ q, and p ∧ q.