1. **Problem:** Construct truth tables for each compound proposition: a) $p \to \neg p$, b) $p \leftrightarrow \neg p$, c) $p \oplus (p \lor q)$, d) $(p \land q) \to (p \lor q)$, e) $(q \to \neg p) \leftrightarrow (p \leftrightarrow q)$, f) $(p \leftrightarrow q) \oplus (p \leftrightarrow \neg q)$. 2. **Explanation of symbols:** - $\neg$: negation (NOT) - $\land$: conjunction (AND) - $\lor$: disjunction (OR) - $\to$: implication - $\leftrightarrow$: biconditional (if and only if) - $\oplus$: exclusive OR (XOR) 3. **Truth table setup:** Since propositions involve $p$ and $q$, list all combinations of truth values: | $p$ | $q$ | |:-:|:-:| | T | T | | T | F | | F | T | | F | F | 4. **Compute each compound:** **a) $p \to \neg p$** - $\neg p$ is negation of $p$ - $p \to \neg p$ is false only when $p$ is true and $\neg p$ false (i.e. $p$ true implies false) | $p$ | $\neg p$ | $p \to \neg p$ | |:-:|:-:|:-:| | T | F | F | | F | T | T | **b) $p \leftrightarrow \neg p$** - True if $p$ and $\neg p$ have same truth value | $p$ | $\neg p$ | $p \leftrightarrow \neg p$ | |:-:|:-:|:-:| | T | F | F | | F | T | F | **c) $p \oplus (p \lor q)$** - $p \lor q$: true if at least one is true - XOR true if operands differ | $p$ | $q$ | $p \lor q$ | $p \oplus (p \lor q)$ | |:-:|:-:|:-:|:-:| | T | T | T | F | | T | F | T | F | | F | T | T | T | | F | F | F | F | **d) $(p \land q) \to (p \lor q)$** - $(p \land q)$ true only if both true - $(p \lor q)$ true if at least one true - Implication true except when antecedent true and consequent false | $p$ | $q$ | $p \land q$ | $p \lor q$ | $(p \land q) \to (p \lor q)$ | |:-:|:-:|:-:|:-:|:-:| | T | T | T | T | T | | T | F | F | T | T | | F | T | F | T | T | | F | F | F | F | T | **e) $(q \to \neg p) \leftrightarrow (p \leftrightarrow q)$** - Compute $q \to \neg p$, $p \leftrightarrow q$, then biconditional | $p$ | $q$ | $\neg p$ | $q \to \neg p$ | $p \leftrightarrow q$ | $(q \to \neg p) \leftrightarrow (p \leftrightarrow q)$ | |:-:|:-:|:-:|:-:|:-:|:-:| | T | T | F | F | T | F | | T | F | F | T | F | F | | F | T | T | T | F | F | | F | F | T | T | T | T | **f) $(p \leftrightarrow q) \oplus (p \leftrightarrow \neg q)$** - Compute $p \leftrightarrow q$ and $p \leftrightarrow \neg q$, then XOR them | $p$ | $q$ | $\neg q$ | $p \leftrightarrow q$ | $p \leftrightarrow \neg q$ | XOR | |:-:|:-:|:-:|:-:|:-:|:-:| | T | T | F | T | F | T | | T | F | T | F | T | T | | F | T | F | F | T | T | | F | F | T | T | F | T | 5. **Final answer:** The truth tables above fully describe the truth values for all compound propositions.