| ( p ) | ( q ) | ( p \land q ) | ( \neg(p \land q) ) | ( \neg p ) | ( \neg q ) | ( \neg p \lor \neg q ) | |--------|--------|----------------|-----------------------|--------------|--------------|--------------------------| | V | V | V | F | F | F | F | | V | F | F | V | F | V | V | | F | V | F | V | V | F | V | | F | F | F | V | V | V | V |

✅ All final values are → Tautology . Exercise 8: Check if Contradiction Problem: Show that ( p \land \neg p ) is a contradiction (always false).

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.

| ( p ) | ( q ) | ( p \lor q ) | |--------|--------|----------------| | V | V | V | | V | F | V | | F | V | V | | F | F | F | Problem: Build the truth table for ( p \to 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 \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 \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 ).