To determine if the two statements $p \to \neg r$ (meaning "if $p$, then not $r$") and $r \lor \neg p$ (meaning "r or not p") are logically equivalent, we will construct a truth table for both and compare their outputs.

Steps:

1. Identify the variables: $p$ and $r$.
2. List all possible truth values for $p$ and $r$.
3. Calculate the truth values for $p \to \neg r$ and $r \lor \neg p$ for each combination of $p$ and $r$.
4. Compare the results.

Truth Table:

$p$	$r$	$\neg r$	$\neg p$	$p \to \neg r$	$r \lor \neg p$
T	T	F	F	F	T
T	F	T	F	T	F
F	T	F	T	T	T
F	F	T	T	T	T

Explanation:

1. Column 1 and 2: All possible truth values of $p$ and $r$ are listed.
2. Column 3 ($\neg r$): Negation of $r$.
3. Column 4 ($\neg p$): Negation of $p$.
4. Column 5 ($p \to \neg r$): A conditional statement is false only when the antecedent $p$ is true and the consequent $\neg r$ is false. Hence, it is false only when $p = T$ and $r = T$.
5. Column 6 ($r \lor \neg p$): Disjunction (OR) is true if at least one of the operands is true. It's only false when both $r = F$ and $\neg p = F$, which happens when $r = F$ and $p = T$.

Conclusion:

The two statements are not logically equivalent because the truth values in columns 5 and 6 differ for some rows (in the second row, for example).

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Would you like a detailed explanation of any of the steps?

Here are five related questions you can explore:

1. What is the logical negation of a compound statement?
2. How can you use truth tables to verify tautologies?
3. Are $p \to q$ and $\neg p \lor q$ logically equivalent?
4. What is the difference between logical equivalence and implication?
5. How do conjunctions and disjunctions behave with truth tables?

Tip: When constructing truth tables, always ensure each logical operation is performed step-by-step to avoid mistakes.