Fill in the truth table for proposition q∨(¬p→q)
T  T
T  F
F  T
F  F

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:

\[
\begin{array}{|c|c|c|c|c|}
\hline
p & q & \neg p & \neg p \rightarrow q & q \vee (\neg p \rightarrow q) \\
\hline
T & T & F & T & T \\
T & F & F & T & T \\
F & T & T & T & T \\
F & F & T & F & F \\
\hline
\end{array}
\]

### 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.

Explore how to fill in the truth table for the proposition q∨(¬p→q) in propositional logic. Understand the logical operations involved and solve step-by-step with detailed explanations. Suitable for college-level students and enthusiasts of logic and mathematics.

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