Construct a truth table for the statement.

r  v (r^~r)

To construct a truth table for the statement \( r \lor (r \land \neg r) \), we need to evaluate the expression step by step:

1. \( r \) - the initial variable.
2. \( \neg r \) - the negation of \( r \).
3. \( r \land \neg r \) - the conjunction (logical AND) of \( r \) and \( \neg r \).
4. \( r \lor (r \land \neg r) \) - the disjunction (logical OR) of \( r \) and the result from step 3.

Now, let's fill in the truth table:

| \( r \) | \( \neg r \) | \( r \land \neg r \) | \( r \lor (r \land \neg r) \) |
|:------:|:-----------:|:------------------:|:----------------------------:|
|   T    |      F      |         F          |              T               |
|   F    |      T      |         F          |              F               |

### Explanation:

- **Column 1 (\( r \))**: Lists the possible truth values of \( r \) (True and False).
- **Column 2 (\( \neg r \))**: Lists the negation of \( r \).
- **Column 3 (\( r \land \neg r \))**: This is always False because \( r \) and \( \neg r \) cannot both be true simultaneously (they are mutually exclusive).
- **Column 4 (\( r \lor (r \land \neg r) \))**: The result is True if \( r \) is True (regardless of \( r \land \neg r \)), and False if \( r \) is False.

Would you like to explore this further or have any questions?

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Here are five related questions you might explore next:

1. What is the truth table for \( r \land (\neg r \lor s) \)?
2. How does the expression \( r \land \neg r \) relate to contradictions?
3. Can you prove that \( r \lor (r \land \neg r) \) simplifies to \( r \)?
4. How do logical operations behave when combining multiple variables, like \( r \lor s \)?
5. What is the importance of the distributive property in logic?

**Tip:** When constructing truth tables, carefully analyze each step, as it helps in understanding the interaction between logical operators.

Learn how to construct a truth table for the logical statement r ∨ (r ∧ ¬r). This problem involves evaluating logical expressions using truth values of variables. Suitable for students in Grades 10-12 studying logic and truth tables.

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