The problem requires completing a truth table for the logical expression $(P \land Q) \rightarrow (Q \lor R)$. We are given columns for $P$, $Q$, $R$, $P \land Q$, $Q \lor R$, and $P \lor \neg R$. We need to fill in the missing values for each row.

1. Problem Description

The problem requires completing a truth table for the logical expression $(P \land Q) \rightarrow (Q \lor R)$. We are given columns for $P$, $Q$, $R$, $P \land Q$, $Q \lor R$, and $P \lor \neg R$. We need to fill in the missing values for each row.

2. Solution Steps

First, let's compute the values for $P \land Q$. The conjunction $P \land Q$ is true only if both $P$ and $Q$ are true.

- Row 1: $T \land T = T$

- Row 2: $T \land T = T$

- Row 3: $T \land F = F$

- Row 4: $T \land F = F$

- Row 5: $F \land T = F$

- Row 6: $F \land T = F$

- Row 7: $F \land F = F$

- Row 8: $F \land F = F$

Next, let's compute the values for $Q \lor R$. The disjunction $Q \lor R$ is true if either $Q$ or $R$ (or both) are true.

- Row 1: $T \lor T = T$

- Row 2: $T \lor F = T$

- Row 3: $F \lor T = T$

- Row 4: $F \lor F = F$

- Row 5: $T \lor T = T$

- Row 6: $T \lor F = T$

- Row 7: $F \lor T = T$

- Row 8: $F \lor F = F$

Now, let's compute the values for $P \lor \neg R$. First, we need to find $\neg R$. $\neg R$ is the negation of $R$.

- Row 1: $\neg T = F$

- Row 2: $\neg F = T$

- Row 3: $\neg T = F$

- Row 4: $\neg F = T$

- Row 5: $\neg T = F$

- Row 6: $\neg F = T$

- Row 7: $\neg T = F$

- Row 8: $\neg F = T$

Now, we can find $P \lor \neg R$. The disjunction $P \lor \neg R$ is true if either $P$ or $\neg R$ (or both) are true.

- 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: $F \lor F = F$

- Row 8: $F \lor T = T$

Finally, we compute the values for $(P \land Q) \rightarrow (Q \lor R)$. The implication $A \rightarrow B$ is false only if $A$ is true and $B$ is false. Otherwise, it is true.

- Row 1: $T \rightarrow T = T$

- Row 2: $T \rightarrow T = T$

- Row 3: $F \rightarrow T = T$

- Row 4: $F \rightarrow F = T$

- Row 5: $F \rightarrow T = T$

- Row 6: $F \rightarrow T = T$

- Row 7: $F \rightarrow T = T$

- Row 8: $F \rightarrow F = T$

3. Final Answer

Here's the completed truth table:

| P | Q | R | P ∧ Q | Q ∨ R | P ∨ ¬R | (P ∧ Q) → (Q ∨ R) |

|---|---|---|-------|-------|-------|-----------------------|

| T | T | T | T | T | T | T |

| T | T | F | T | T | T | T |

| T | F | T | F | T | T | T |

| T | F | F | F | F | T | T |

| F | T | T | F | T | F | T |

| F | T | F | F | T | T | T |

| F | F | T | F | T | F | T |

| F | F | F | F | F | T | T |