Construct a truth table for the statement to determine if the statements are equivalent.
p → ~r; r ∨ ~p

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.

Construct a truth table for the statement to determine if the statements are equivalent: p → ~r; r ∨ ~p.

Construct a truth table to determine the logical equivalence of p → ¬r and r ∨ ¬p. This problem explores key concepts in propositional logic, focusing on conditional statements and disjunctions, with a step-by-step explanation of how to compare truth values. Suitable for Grades 10-12.

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