Problem:
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an on the exam. Which one of these statements necessarily follows logically?
Answer Choices:
A. If Lewis did not receive an A, then he got all of the multiple choice questions wrong.
B. If Lewis did not receive an , then he got at least one of the multiple choice questions wrong.
C. If Lewis got at least one of the multiple choice questions wrong, then he did not receive an .
D. If Lewis received an A, then he got all of the multiple choice questions right.
E. If Lewis received an , then he got at least one of the multiple choice questions right.
Solution:
The given statement is logically equivalent to its contrapositive: If a student did not receive an on the exam, then the student did not get all the multiple choice questions right, which means that he got at least one of them wrong. None of the other statements follows logically from the given implication; the teacher made no promises concerning students who did not get all the multiple choice questions right. In particular, a statement does not imply its inverse or its converse; and the negation of the statement that Lewis got all the questions right is not the statement that he got all the questions wrong.
The problems on this page are the property of the MAA's American Mathematics Competitions