Let's break down the truth table and complete it step-by-step for the expression:

$((p \leftrightarrow q) \land \neg q) \rightarrow \neg p$

We'll evaluate the components in a systematic way.

Step 1: Recall the columns of the truth table provided.

1. $p$ and $q$: These represent the truth values of the variables $p$ and $q$.
2. $\neg q$ and $\neg p$: These are the negations of $q$ and $p$, respectively.
3. $p \leftrightarrow q$: The biconditional statement, which is true if both $p$ and $q$ have the same truth values, and false otherwise.
4. $(p \leftrightarrow q) \land \neg q$: The conjunction of the biconditional and the negation of $q$.
5. The final expression: $((p \leftrightarrow q) \land \neg q) \rightarrow \neg p$.

Step 2: Truth Table Structure

We will go row by row and calculate each of these components for the four possible truth value combinations of $p$ and $q$.

Step 3: Explanation

• Biconditional $p \leftrightarrow q$:
  • True only when $p$ and $q$ are both true or both false.
• Conjunction $(p \leftrightarrow q) \land \neg q$:
  • True only if the biconditional is true and $\neg q$ is also true.
• Implication $((p \leftrightarrow q) \land \neg q) \rightarrow \neg p$:
  • The implication is true except in the case where the premise is true and the conclusion is false.

Thus, all rows evaluate to true for the final expression.

Would you like more details on any of these steps or have further questions? Here are 5 related questions to expand on this topic:

1. How do truth tables help in understanding logical expressions?
2. What is the difference between conjunction and biconditional operations?
3. How does the truth value of an implication change based on its components?
4. What happens if you change the order of operations in a logical expression?
5. How can you use a truth table to determine if a logical expression is a tautology?

Tip: When dealing with implications, remember that the statement is only false when the premise is true and the conclusion is false.