To determine if the given statement $(p \vee q) \vee (\neg p \vee q)$ is a tautology using a truth table, we need to analyze all possible truth values of $p$ and $q$.

Step-by-step:

1. List all possible truth values for $p$ and $q$.
2. Compute intermediate expressions: $p \vee q$, $\neg p$, $\neg p \vee q$, and finally $(p \vee q) \vee (\neg p \vee q)$.
3. Check if the final column (the result of the whole expression) is always true.

Truth Table:

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

Explanation:

• Column $p \vee q$ shows the result of $p$ OR $q$.
• Column $\neg p$ shows the negation of $p$.
• Column $\neg p \vee q$ shows the result of $\neg p$ OR $q$.
• Finally, $(p \vee q) \vee (\neg p \vee q)$ is computed by taking the OR of the two previous columns.

Conclusion:

The last column shows that the statement $(p \vee q) \vee (\neg p \vee q)$ is true for all possible truth values of $p$ and $q$. Therefore, the given statement is a tautology.

Would you like further clarification or details on any part of this process?

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Here are 5 related questions:

1. How can you identify a tautology without using a truth table?
2. What is the difference between a tautology and a contradiction in logic?
3. Can a statement with more than two variables be a tautology?
4. How does the distributive property of logical operators apply to this problem?
5. What are common tautological forms in propositional logic?

Tip: When constructing a truth table, always double-check intermediate expressions to avoid mistakes when dealing with more complex statements.