The task is to determine whether the collection of logical expressions is satisfiable based on the given truth table. Let's break down the logical expressions and see if we can complete the table.

The expressions are:

1. $\neg q \lor (p \rightarrow q)$
2. $(\neg p) \rightarrow q$
3. $\neg p \land (q \lor p)$

Step-by-step evaluation:

We will evaluate the expressions for the missing truth values and then determine if there is a row where all expressions are true.

Logical notation:

• $p \rightarrow q$ is true unless $p$ is true and $q$ is false.
• $\neg$ is the negation (NOT operator).
• $\lor$ is logical OR.
• $\land$ is logical AND.

Row-by-row evaluation:

Row 3: $p = F, q = T$

• $\neg q \lor (p \rightarrow q)$:
  • $\neg q = F$
  • $p \rightarrow q = T$ (since $p = F$, the implication is always true)
  • $F \lor T = T$
• $(\neg p) \rightarrow q$:
  • $\neg p = T$
  • $T \rightarrow q = T$ (since $q = T$)
  • So, this is true.
• $\neg p \land (q \lor p)$:
  • $\neg p = T$, $q \lor p = T$
  • $T \land T = T$

Row 3 satisfies all conditions.

Row 4: $p = F, q = F$

• $\neg q \lor (p \rightarrow q)$:
  • $\neg q = T$
  • $p \rightarrow q = T$ (since $p = F$)
  • $T \lor T = T$
• $(\neg p) \rightarrow q$:
  • $\neg p = T$
  • $T \rightarrow F = F$
• $\neg p \land (q \lor p)$:
  • $\neg p = T$
  • $q \lor p = F$
  • $T \land F = F$

Row 4 does not satisfy all conditions.

Conclusion:

The collection is satisfiable, and Row 3 is the one that satisfies all expressions.

Answers:

1. Is the collection satisfiable? Yes
2. Enter a row number that satisfies the collection: 3

Would you like more details on this solution or further clarification?

Here are 5 related questions to consider:

1. What is the difference between logical conjunction (AND) and disjunction (OR)?
2. How does the truth value of an implication ($p \rightarrow q$) work?
3. Why is Row 4 unsatisfiable for the expression $(\neg p) \rightarrow q$?
4. Can a logical collection be satisfiable with multiple rows satisfying all conditions?
5. How does negation affect the truth values of complex expressions?

Tip: When dealing with logical expressions, it helps to break them down into smaller parts and evaluate step-by-step to avoid confusion.