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.