AMC10 2017 A

AMC10 2017 A · Q6

AMC10 2017 A · Q6. It mainly tests Logic puzzles.

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?

卡罗尔女士承诺，任何在即将到来的考试中全部答对选择题的人都将获得考试A等。以下哪个陈述必然逻辑上成立？

(A) If Lewis did not receive an A, then he got all of the multiple choice questions wrong. 如果Lewis没有得到A，那么他所有选择题都错了。

(B) If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. 如果Lewis没有得到A，那么他至少有一道选择题错了。

(C) If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. 如果Lewis至少有一道选择题错了，那么他没有得到A。

(D) If Lewis received an A, then he got all of the multiple choice questions right. 如果Lewis得到了A，那么他所有选择题都对了。

(E) If Lewis received an A, then he got at least one of the multiple choice questions right. 如果Lewis得到了A，那么他至少有一道选择题对了。

Answer

Correct choice: (B)

正确答案：（B）

Solution

The given statement is logically equivalent to its contrapositive: If a student did not receive an A 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.

给定的陈述逻辑上等价于其逆否命题：如果一个学生没有在考试中获得A等，那么该学生没有全部答对选择题，这意味着他至少错了一题。其他陈述都不从给定的蕴涵中逻辑上得出；老师没有对没有全部答对选择题的学生做出任何承诺。特别是，一个陈述并不蕴涵其逆命题或其逆否命题；Lewis全部答对题目的否定不是他全部答错的陈述。

Topics

LM.001 Logic puzzles