AMC12 2017 A

AMC12 2017 A · Q3

AMC12 2017 A · Q3. 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?

Carroll 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，则他所有多项选择题都错了。

(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，则他至少有一道多项选择题错了。

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

(D) If Lewis received an A, then he got all of the multiple choice questions right. 如果 Lewis 得到 A，则他所有多项选择题都对了。

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

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): 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.

答案（B）：给定陈述在逻辑上等价于它的逆否命题：如果一个学生在考试中没有得到 A，那么该学生就没有把所有选择题都答对，这意味着他至少有一道题答错了。其他任何陈述都不能从给定的蕴含关系中逻辑地推出；老师并没有对那些没有把所有选择题都答对的学生作出任何承诺。特别地，一个陈述并不蕴含它的否命题或逆命题；并且，“刘易斯把所有题都答对了”这一陈述的否定，并不是“他把所有题都答错了”。

Topics

LM.001 Logic puzzles