To fill in the truth table for the proposition $q \vee (\neg p \rightarrow q)$, we need to first determine the truth value of the implication $\neg p \rightarrow q$, and then compute the disjunction $q \vee (\neg p \rightarrow q)$.

The implication $\neg p \rightarrow q$ is true except when $\neg p$ is true and $q$ is false.

Let's fill in the truth table step by step:

1. Negation $\neg p$: Determine the truth value of $\neg p$.
2. Implication $\neg p \rightarrow q$: Determine the truth value of the implication based on $\neg p$ and $q$.
3. Disjunction $q \vee (\neg p \rightarrow q)$: Determine the final truth value by combining $q$ with $\neg p \rightarrow q$ using the OR operation.

Here is the completed truth table:

\hline p & q & \neg p & \neg p \rightarrow q & q \vee (\neg p \rightarrow q) \\ \hline T & T & F & T & T \\ T & F & F & T & T \\ F & T & T & T & T \\ F & F & T & F & F \\ \hline \end{array}$$ ### Explanation: - **First row:** $$p = T$$, $$q = T$$, $$\neg p = F$$, $$\neg p \rightarrow q = T$$ (since $$\neg p$$ is false), so $$q \vee T = T$$. - **Second row:** $$p = T$$, $$q = F$$, $$\neg p = F$$, $$\neg p \rightarrow q = T$$ (since $$\neg p$$ is false), so $$q \vee T = T$$. - **Third row:** $$p = F$$, $$q = T$$, $$\neg p = T$$, $$\neg p \rightarrow q = T$$ (since both are true), so $$q \vee T = T$$. - **Fourth row:** $$p = F$$, $$q = F$$, $$\neg p = T$$, $$\neg p \rightarrow q = F$$ (since $$\neg p$$ is true and $$q$$ is false), so $$q \vee F = F$$. Would you like further details on any step or have other questions? Here are 5 related questions for further practice: 1. What is the truth table for $$\neg p \vee (\neg p \rightarrow q)$$? 2. How does $$p \rightarrow q$$ compare to $$\neg p \rightarrow q$$? 3. Can you create a truth table for $$(q \vee \neg p) \rightarrow q$$? 4. What is the truth table for $$q \vee (\neg p \wedge q)$$? 5. How would the truth table change for $$q \vee (\neg p \rightarrow \neg q)$$? **Tip:** For implications ($$\rightarrow$$), the only time the statement is false is when the first part (antecedent) is true and the second part (consequent) is false.