Use a truth table to determine whether the given statement is a tautology.
(p ∨ q) ∨ (~p ∨ q)

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.

Use a truth table to determine whether the given statement is a tautology. (p ∨ q) ∨ (~p ∨ q)

Use a truth table to analyze whether the logical statement (p ∨ q) ∨ (¬p ∨ q) is a tautology. This problem covers key concepts in propositional logic, including truth tables, logical connectives, and the definition of a tautology. Suitable for students in Grades 10-12.

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