1. **Problem statement:** Determine the validity of each argument and identify the rule of inference or logical error. 2. **Recall important rules:** - **Conditional statement:** If $p$, then $q$ ($p \to q$). - **Converse:** If $q$, then $p$ (not necessarily true). - **Contrapositive:** If not $q$, then not $p$ (always true). - **Inverse:** If not $p$, then not $q$ (not necessarily true). 3. **Analyze each argument:** **a)** Given: If $n > 1$, then $n^2 > 1$. - Argument: Suppose $n^2 > 1$. Then $n > 1$. - This is the converse of the original statement. - The converse is not always true (e.g., $n = -2$ gives $n^2 = 4 > 1$ but $n \not> 1$). - **Conclusion:** Argument is invalid; logical error is affirming the consequent. **b)** Given: If $n > 3$, then $n^2 > 9$. - Argument: Suppose $n^2 \leq 9$. Then $n \leq 3$. - This is the contrapositive of the original statement. - Contrapositive is logically equivalent to the original statement. - **Conclusion:** Argument is valid; rule of inference is contrapositive. **c)** Given: If $n > 2$, then $n^2 > 4$. - Argument: Suppose $n \leq 2$. Then $n^2 \leq 4$. - This is the inverse of the original statement. - The inverse is not necessarily true (e.g., $n = -3$ gives $n \leq 2$ but $n^2 = 9 \not\leq 4$). - **Conclusion:** Argument is invalid; logical error is denying the antecedent. 4. **Summary:** - a) Invalid, affirming the consequent. - b) Valid, contrapositive. - c) Invalid, denying the antecedent.