To construct a truth table for the expression $p \land (q \lor \sim r)$, let's break down the steps and create a table that examines all possible truth values of $p$, $q$, and $r$.

Variables and Operators

• $p$: First variable, which can be either True (T) or False (F).
• $q$: Second variable, also True or False.
• $r$: Third variable, also True or False.
• $\sim r$: The negation of $r$.
• $q \lor \sim r$: The logical "OR" of $q$ and $\sim r$.
• $p \land (q \lor \sim r)$: The logical "AND" of $p$ and $(q \lor \sim r)$.

Truth Table

Here’s the step-by-step truth table calculation for each logical operation.

Explanation of Columns

1. $p$, $q$, $r$: These columns show each possible combination of truth values for the variables $p$, $q$, and $r$.
2. $\sim r$: This column is the negation of $r$.
3. $q \lor \sim r$: This column shows the result of the "OR" operation between $q$ and $\sim r$.
4. $p \land (q \lor \sim r)$: This is the final column, showing the result of the "AND" operation between $p$ and $(q \lor \sim r)$.

Would you like further details on any part of the process or have any questions?

Additional Questions

1. How is the negation operator $\sim$ applied in logical statements?
2. What is the difference between "AND" ($\land$) and "OR" ($\lor$) in logic?
3. How would the truth table change if the expression was $p \lor (q \land \sim r)$?
4. Can truth tables be used to verify logical equivalences?
5. What are common applications of truth tables in computer science and logic?

Tip:

In truth tables, make sure each possible combination of variables is represented; for three variables, there are always $2^3 = 8$ rows.