1975 AMC 12 Problem 8

Question

If the statement “All shirts in this store are on sale.” is false, then which of the following statements must be true?

I. All shirts in this store are at non-sale prices.

II. There is some shirt in this store not on sale.

III. No shirt in this store is on sale.

IV. Not all shirts in this store are on sale.

Note: Originally, statement I read: All shirts in this store are not on sale.

\mathrm{II}

only

\mathrm{IV}

only

\mathrm{I}

and

\mathrm{III}

only

\mathrm{II}

and

\mathrm{IV}

only

\mathrm{I},

\mathrm{II}

and

\mathrm{IV}

only

Answer: D

Concepts:logical deduction

Difficulty rating: 1380

Small hint:

Negating “all” asserts the existence of a counterexample

Big hint:

The false statement does not say that every shirt fails to be on sale

Solution:

The negation of “Every shirt is on sale” is “Some shirt is not on sale,” which is II and is equivalently phrased by IV. Statements I and III make the stronger claim that no shirt is on sale, which need not follow.

Therefore, the correct answer is D.

1975 AMC 12 Problem 8

Answer: D

Difficulty rating: 1380

Small hint:

Big hint:

Solution:

Problem 8 in other years