Construct a truth table for the statement.

r  (r  ~r)

To construct a truth table for the statement \( r \land (r \lor \neg r) \), we need to consider all possible truth values for the variable \( r \) and then evaluate each part of the expression step by step.

Here's the breakdown:

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

Now, let's fill in the truth table:

| \( r \) | \( \neg r \) | \( r \lor \neg r \) | \( r \land (r \lor \neg r) \) |
|:------:|:-----------:|:------------------:|:-----------------------------:|
|   T    |      F      |         T          |              T                |
|   F    |      T      |         T          |              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 (\( r \lor \neg r \))**: This is always True because \( r \) or \( \neg r \) covers all possibilities (one of them will always be true).
- **Column 4 (\( r \land (r \lor \neg r) \))**: The result depends on \( r \). If \( r \) is True, the result is True; if \( r \) is False, the result is False.

Would you like to go into more detail or have any questions?

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Here are five related questions you might explore next:

1. What is the truth table for \( \neg(r \land s) \)?
2. How does De Morgan's Law apply to logical expressions?
3. Can you create a truth table for \( r \land s \lor \neg t \)?
4. What is the difference between a tautology and a contradiction?
5. How do truth tables help in understanding logical equivalences?

**Tip:** Practice constructing truth tables for different logical expressions to get a better grasp of how each logical operator works.

Learn how to construct a truth table for the logical expression r ∧ (r ∨ ¬r). This problem explores fundamental concepts in logic and truth tables, covering conjunctions, disjunctions, and negations. Suitable for high school students studying logic and Boolean algebra.

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