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| ( p ) | ( q ) | ( r ) | ( p \lor q ) | ( \neg r ) | ( (p \lor q) \to \neg r ) | |--------|--------|--------|----------------|--------------|-----------------------------| | V | V | V | V | F | F | | V | V | F | V | V | V | | V | F | V | V | F | F | | V | F | F | V | V | V | | F | V | V | V | F | F | | F | V | F | V | V | V | | F | F | V | F | F | V (F → F = V) | | F | F | F | F | V | V (F → V = V) | Problem: Show that ( (p \to q) \lor (q \to p) ) is a tautology (always true).
| ( p ) | ( q ) | ( p \to q ) | ( q \to p ) | ( (p \to q) \lor (q \to p) ) | |--------|--------|--------------|--------------|-------------------------------| | V | V | V | V | V | | V | F | F | V | V | | F | V | V | F | V | | F | F | V | V | V |
1. Introduction Mathematical logic is the foundation of all reasoning in mathematics and computer science. A truth table is a systematic way to list all possible truth values (True or False, often denoted as ( V ) or ( F ), or ( 1 ) and ( 0 )) of a logical proposition based on the truth values of its components. --- Logica Matematica Tablas De Verdad Ejercicios Resueltos
| ( p ) | ( q ) | ( p \to q ) | |--------|--------|----------------| | V | V | V | | V | F | F | | F | V | V | | F | F | V | An implication is only false when the antecedent ((p)) is true and the consequent ((q)) is false. Exercise 5: Biconditional Problem: Build the truth table for ( p \leftrightarrow q ).
| ( p ) | ( q ) | ( p \land q ) | |--------|--------|----------------| | V | V | V | | V | F | F | | F | V | F | | F | F | F | Problem: Build the truth table for ( p \lor q ). | ( p ) | ( q )
| ( p ) | ( q ) | ( p \leftrightarrow q ) | |--------|--------|---------------------------| | V | V | V | | V | F | F | | F | V | F | | F | F | V | Problem: Build the truth table for ( (p \lor q) \to \neg r ).
( p, q, r ) → ( 2^3 = 8 ) rows.
✅ All final values are → Contradiction . Exercise 9: Logical Equivalence Problem: Verify that ( \neg (p \land q) \equiv \neg p \lor \neg q ) (De Morgan’s Law).