1. The problem asks to identify whether $[(p \to q) \lor (q \to r) \lor (r \to p)]$ is a tautology, contradiction, or contingency. 2. The problem also asks to identify whether $[((p \lor q) \lor r) \land ((\neg p \land \neg q) \land \neg r)]$ is a tautology, contradiction, or contingency. **Step 1**: Recall the definitions: - A **tautology** is true in every possible truth assignment. - A **contradiction** is false in every possible truth assignment. - A **contingency** is true in some truth assignments and false in others. **Step 2**: Analyze statement 1: $[(p \to q) \lor (q \to r) \lor (r \to p)]$ - Implication $p \to q$ is false only if $p$ is true and $q$ is false. - We want to check if this disjunction is true for all truth assignments of $p,q,r$. Construct truth table variables (values of $p,q,r$) and determine $p \to q$, $q \to r$, and $r \to p$; then their disjunction: | $p$ | $q$ | $r$ | $p \to q$ | $q \to r$ | $r \to p$ | Disjunction | |-----|-----|-----|-----------|-----------|-----------|-------------| | T | T | T | T | T | T | T | | T | T | F | T | F | F | T | | T | F | T | F | T | T | T | | T | F | F | F | T | F | T | | F | T | T | T | T | T | T | | F | T | F | T | F | T | T | | F | F | T | T | T | T | T | | F | F | F | T | T | T | T | Since the disjunction is true in all cases, statement 1 is a **tautology**. **Step 3**: Analyze statement 2: $[((p \lor q) \lor r) \land ((\neg p \land \neg q) \land \neg r)]$ Breaking down: - Left part $(p \lor q) \lor r$ is true if at least one of $p,q,r$ is true. - Right part $(\neg p \land \neg q) \land \neg r$ means all $p,q,r$ are false. We have a conjunction of these parts, so both must be true simultaneously. Check if possible: - Left: at least one true - Right: all false This is impossible. Hence statement 2 is always false: a **contradiction**. **Final answer:** 1. $[(p \to q) \lor (q \to r) \lor (r \to p)]$ is a tautology. 2. $[((p \lor q) \lor r) \land ((\neg p \land \neg q) \land \neg r)]$ is a contradiction.