To create the truth table for the statement $(p \vee q) \land (\neg p \vee \neg r)$, we'll follow these steps:

Variables

We have three variables: $p, q, r$. Therefore, there are $2^3 = 8$ combinations of truth values for these variables.

Logical Symbols

• $\vee$: OR
• $\land$: AND
• $\neg$: NOT

Truth Table Construction

$p$	$q$	$r$	$\neg p$	$\neg r$	$p \vee q$	$\neg p \vee \neg r$	$(p \vee q) \land (\neg p \vee \neg r)$
T	T	T	F	F	T	F	F
T	T	F	F	T	T	T	T
T	F	T	F	F	T	F	F
T	F	F	F	T	T	T	T
F	T	T	T	F	T	T	T
F	T	F	T	T	T	T	T
F	F	T	T	F	F	T	F
F	F	F	T	T	F	T	F

Explanation

1. Columns 1-3: Assign all possible combinations of truth values to $p, q, r$.
2. Column 4 ($\neg p$): The negation of $p$.
3. Column 5 ($\neg r$): The negation of $r$.
4. Column 6 ($p \vee q$): The OR operation between $p$ and $q$.
5. Column 7 ($\neg p \vee \neg r$): The OR operation between $\neg p$ and $\neg r$.
6. Final Column: The AND operation between $(p \vee q)$ and $(\neg p \vee \neg r)$.

This completes the truth table for the statement $(p \vee q) \land (\neg p \vee \neg r)$.

Would you like further details or have any questions about this?

Related Questions:

1. How do you interpret the logical operators AND, OR, and NOT in a truth table?
2. What would happen if you change the AND ($\land$) operator to an OR ($\vee$) in the statement?
3. How does the number of variables affect the size of a truth table?
4. How can truth tables help in verifying logical equivalences?
5. What is the significance of using De Morgan’s laws in simplifying logical statements?

Tip: In a truth table, always start by assigning truth values to each variable, and then build the table step by step for each component of the logical expression.