1. We are asked to construct truth tables for four compound propositions involving variables $p$ and $q$. 2. Recall the logical operators: - $\neg$ means NOT (negation). - $\to$ means IMPLIES. - $\leftrightarrow$ means BICONDITIONAL (if and only if). - $\vee$ means OR. 3. List all combinations of $p$ and $q$: $p:\{T,F\}$ and $q:\{T,F\}$ gives 4 rows: $(T,T),(T,F),(F,T),(F,F)$. **a) Truth table for $p \to \neg q$**: - Compute $\neg q$. - Then $p \to \neg q$ is false only if $p$ is true and $\neg q$ is false. | $p$ | $q$ | $\neg q$ | $p \to \neg q$ | |-----|-----|----------|-----------------| | T | T | F | F | | T | F | T | T | | F | T | F | T | | F | F | T | T | **b) Truth table for $\neg p \leftrightarrow q$**: - Compute $\neg p$. - Biconditional is true if both values are the same. | $p$ | $q$ | $\neg p$ | $\neg p \leftrightarrow q$ | |-----|-----|----------|-----------------------------| | T | T | F | F | | T | F | F | T | | F | T | T | T | | F | F | T | F | **c) Truth table for $(p \to q) \vee (\neg p \to q)$**: - Compute $p \to q$ and $\neg p \to q$. - OR the two values. | $p$ | $q$ | $p \to q$ | $\neg p$ | $\neg p \to q$ | $(p \to q) \vee (\neg p \to q)$ | |-----|-----|------------|----------|-----------------|-----------------------------------| | T | T | T | F | T | T | | T | F | F | F | T | T | | F | T | T | T | T | T | | F | F | T | T | F | T | **d) Truth table for $(p \to q)$**: | $p$ | $q$ | $p \to q$ | |-----|-----|-----------| | T | T | T | | T | F | F | | F | T | T | | F | F | T | Summary: All truth tables have been created showing intermediate steps clearly.