1976 AHSME Problems/Problem 11

Problem 11

Which of the following statements is (are) equivalent to the statement "If the pink elephant on planet alpha has purple eyes, then the wild pig on planet beta does not have a long nose"?

$\begin{array}{l}\textbf{I. }\text{"If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha has purple eyes."}\\ \textbf{II. }\text{"If the pink elephant on planet alpha does not have purple eyes, then the wild pig on planet beta does not have a long nose."}\\ \textbf{III. }\text{"If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha does not have purple eyes."}\\ \textbf{IV. }\text{"The pink elephant on planet alpha does not have purple eyes, or the wild pig on planet beta does not have a long nose."}\end{array}$

$\textbf{(A) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad \textbf{(B) }\textbf{III. }\text{and }\textbf{IV. }\text{only}\qquad \textbf{(C) }\textbf{II. }\text{and }\textbf{IV. }\text{only}\qquad \textbf{(D) }\textbf{II. }\text{and }\textbf{III. }\text{only}\qquad \textbf{(E) }\textbf{III. }\text{only}$

Solution

The following statements are all logically equivalent.

I. P implies Q (pink elephant on planet alpha has purple eyes ---> wild pig on planet beta does not have a long nose)

III. Not Q implies not P (contrapositive)

IV. Not P nor Q (inverse)

I is the given statement, so III and IV work. The answer is $\boxed{\textbf{(B)}}.$

1976 AHSME (Problems • Answer Key • Resources)
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1976 AHSME Problems/Problem 11

Problem 11

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