Truth Table B993Eb

1. **State the problem:** Construct the truth table for the logical expression $$ (q \to p) \lor \sim r $$ where $p$, $q$, and $r$ are boolean variables. 2. **Recall definitions:** - The implication $q \to p$ is false only when $q$ is true and $p$ is false; otherwise, it is true. - The negation $\sim r$ is true when $r$ is false, and false when $r$ is true. - The disjunction $\lor$ is true if at least one operand is true. 3. **List all possible truth values for $p$, $q$, and $r$: (8 rows)** | $p$ | $q$ | $r$ | |---|---|---| | T | T | T | | T | T | F | | T | F | T | | T | F | F | | F | T | T | | F | T | F | | F | F | T | | F | F | F | 4. **Calculate $q \to p$ for each row:** - Row 1: $q=T$, $p=T$ so $q \to p = T$ - Row 2: $q=T$, $p=T$ so $q \to p = T$ - Row 3: $q=F$, $p=T$ so $q \to p = T$ (implication true if $q$ false) - Row 4: $q=F$, $p=T$ so $q \to p = T$ - Row 5: $q=T$, $p=F$ so $q \to p = F$ - Row 6: $q=T$, $p=F$ so $q \to p = F$ - Row 7: $q=F$, $p=F$ so $q \to p = T$ - Row 8: $q=F$, $p=F$ so $q \to p = T$ 5. **Calculate $\sim r$ for each row:** - Row 1: $r=T$ so $\sim r = F$ - Row 2: $r=F$ so $\sim r = T$ - Row 3: $r=T$ so $\sim r = F$ - Row 4: $r=F$ so $\sim r = T$ - Row 5: $r=T$ so $\sim r = F$ - Row 6: $r=F$ so $\sim r = T$ - Row 7: $r=T$ so $\sim r = F$ - Row 8: $r=F$ so $\sim r = T$ 6. **Calculate $(q \to p) \lor \sim r$ for each row:** - Row 1: $T \lor F = T$ - Row 2: $T \lor T = T$ - Row 3: $T \lor F = T$ - Row 4: $T \lor T = T$ - Row 5: $F \lor F = F$ - Row 6: $F \lor T = T$ - Row 7: $T \lor F = T$ - Row 8: $T \lor T = T$ 7. **Final truth table:** | $p$ | $q$ | $r$ | $(q \to p) \lor \sim r$ | |---|---|---|---| | T | T | T | T | | T | T | F | T | | T | F | T | T | | T | F | F | T | | F | T | T | F | | F | T | F | T | | F | F | T | T | | F | F | F | T |