1. **State the problem:** Construct a truth table for the compound proposition $$((\neg p \lor r) \land (p \leftrightarrow q)) \to r$$. 2. **Recall the logical operators:** - Negation: $$\neg p$$ is true when $$p$$ is false. - Disjunction: $$p \lor q$$ is true if at least one of $$p$$ or $$q$$ is true. - Conjunction: $$p \land q$$ is true only if both $$p$$ and $$q$$ are true. - Biconditional: $$p \leftrightarrow q$$ is true if $$p$$ and $$q$$ have the same truth value. - Implication: $$p \to q$$ is false only if $$p$$ is true and $$q$$ is false; otherwise true. 3. **List all possible truth values for $$p$$, $$q$$, and $$r$$:** There are 8 combinations. | $$p$$ | $$q$$ | $$r$$ | $$\neg p$$ | $$\neg p \lor r$$ | $$p \leftrightarrow q$$ | $$((\neg p \lor r) \land (p \leftrightarrow q))$$ | $$((\neg p \lor r) \land (p \leftrightarrow q)) \to r$$ | |-------|-------|-------|------------|------------------|---------------------|---------------------------------------------|---------------------------------------------| | T | T | T | F | T | T | T | T | | T | T | F | F | F | T | F | T | | T | F | T | F | T | F | F | T | | T | F | F | F | F | F | F | T | | F | T | T | T | T | F | F | T | | F | T | F | T | T | F | F | T | | F | F | T | T | T | T | T | T | | F | F | F | T | T | T | T | F | 4. **Interpretation:** The truth table shows the truth value of the compound proposition for all possible inputs. Final answer: The truth table is constructed as above.