1. The problem involves constructing truth tables for the given logical expressions: 2. The expressions are: 1) $(x \wedge y) \lor z$ 2) $x \to (z \leftrightarrow y)$ 3) $x \lor (z \wedge y)$ 4) $z \to (x \lor y)$ 5) $(x \lor y) \leftrightarrow z$ 3. Important rules: - $\wedge$ means AND: true if both operands are true. - $\lor$ means OR: true if at least one operand is true. - $\to$ means implication: false only if the first is true and the second is false. - $\leftrightarrow$ means biconditional: true if both operands have the same truth value. 4. We list all possible truth values for $x$, $y$, and $z$ (each can be true (T) or false (F)): There are $2^3=8$ combinations. 5. Construct the truth table step-by-step for the first expression $(x \wedge y) \lor z$: | $x$ | $y$ | $z$ | $x \wedge y$ | $(x \wedge y) \lor z$ | |-----|-----|-----|--------------|-----------------------| | F | F | F | F | F | | F | F | T | F | T | | F | T | F | F | F | | F | T | T | F | T | | T | F | F | F | F | | T | F | T | F | T | | T | T | F | T | T | | T | T | T | T | T | 6. Similarly, truth tables can be constructed for the other expressions using the same approach. Final answer: The truth table for the first expression $(x \wedge y) \lor z$ is shown above. This completes the solution for the first problem.