1. **Problem Statement:** Construct truth tables for the compound propositions: a) $p \oplus p$ b) $p \oplus \neg p$ c) $p \oplus \neg q$ d) $\neg p \oplus \neg q$ e) $(p \oplus q) \lor (p \oplus \neg q)$ f) $(p \oplus q) \land (p \oplus \neg q)$ 2. **Recall:** The exclusive or (XOR) operator $\oplus$ is true when exactly one operand is true, false otherwise. 3. **Construct truth tables:** | $p$ | $q$ | $\neg p$ | $\neg q$ | $p \oplus p$ | $p \oplus \neg p$ | $p \oplus \neg q$ | $\neg p \oplus \neg q$ | $p \oplus q$ | $(p \oplus q) \lor (p \oplus \neg q)$ | $(p \oplus q) \land (p \oplus \neg q)$ | |-----|-----|----------|----------|--------------|--------------------|--------------------|-------------------------|--------------|---------------------------------------|---------------------------------------| | T | T | F | F | F | F | T | T | F | T | F | | T | F | F | T | F | T | F | F | T | T | F | | F | T | T | F | F | T | T | T | T | T | T | | F | F | T | T | F | F | F | F | F | F | F | 4. **Explanation:** - For a), $p \oplus p$ is always false because XOR of a value with itself is false. - For b), $p \oplus \neg p$ is always true because $p$ and $\neg p$ are always opposite. - For c) and d), evaluate XOR between $p$ and $\neg q$, and between $\neg p$ and $\neg q$ respectively. - For e), OR the results of $p \oplus q$ and $p \oplus \neg q$. - For f), AND the results of $p \oplus q$ and $p \oplus \neg q$. 5. **Final answers:** - a) $p \oplus p$ is always false. - b) $p \oplus \neg p$ is always true. - c) $p \oplus \neg q$ varies as shown. - d) $\neg p \oplus \neg q$ varies as shown. - e) $(p \oplus q) \lor (p \oplus \neg q)$ is true except when both are false. - f) $(p \oplus q) \land (p \oplus \neg q)$ is true only when both are true.