1976 AMC 12 Problem 11

Question

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”?

\mathrm{I}.

“If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha has purple eyes.”

\mathrm{II}.

“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.”

\mathrm{III}.

“If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha does not have purple eyes.”

\mathrm{IV}.

“The pink elephant on planet alpha does not have purple eyes, or the wild pig on planet beta does not have a long nose.”

The word “or” is used here in the inclusive sense (as is customary in mathematical writing).

\mathrm{I}

and

\mathrm{III}

only

\mathrm{III}

and

\mathrm{IV}

only

\mathrm{II}

and

\mathrm{IV}

only

\mathrm{II}

and

\mathrm{III}

only

\mathrm{III}

only

Answer: B

Concepts:logical deduction

Difficulty rating: 1740

Small hint:

Write the original implication as

P\mathbin{\Rightarrow}\neg Q

Big hint:

Use both its contrapositive and the equivalent disjunction

\neg P\mathbin{\lor}\neg Q

Solution:

Let

P

mean the elephant has purple eyes and

Q

mean the pig has a long nose. The original statement

P\Rightarrow\neg Q

is equivalent to its contrapositive

Q\Rightarrow\neg P,

which is

\mathrm{III},

and to

\neg P\lor\neg Q,

which is

\mathrm{IV}.

Statements

\mathrm{I}

and

\mathrm{II}

are converses and need not follow.

Therefore, the correct answer is B.

1976 AMC 12 Problem 11

Answer: B

Difficulty rating: 1740

Small hint:

Big hint:

Solution:

Problem 11 in other years