题目信息
The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ?
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B:Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D:EACH statement ALONE is sufficient.
E:Statements (1) and (2) TOGETHER are NOT sufficient.
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正确答案：
D:EACH statement ALONE is sufficient.
Arithmetic Concepts of sets
Let n be the cardinality of the finite set A. What is the value of n ?
The number of 2-element subsets of A is equal to the number of unordered selections without replacement of 2 objects from a collection of n distinct objects, or “n choose 2.” Therefore, we have
= 6, or equivalently, n2 – n – 12 = 0. Because this is a quadratic equation that may have two solutions, we need to investigate further to determine whether there is a unique value of n. Factoring leads to (n – 4)(n + 3) = 0, and thus n = 4 or n = −3. Since n must be a nonnegative integer, we discard the solution n = −3. Therefore, n = 4; SUFFICIENT.
The number of subsets of set A is 2n, because each subset of A corresponds to a unique sequence of answers to yes-no questions about membership in the subset (one question for each of the n elements). For example, let A = {1, 2, 3, 4, 5}, let Y represent “yes,” and let N represent “no.” Then the sequence NYNNN corresponds to the subset {2}, since the answers to “is 1 in the subset,” “is 2 in the subset,” “is 3 in the subset,” etc. are “no,” “yes,” “no,” etc. Also, the subset {1, 3, 4} of A corresponds to the 5-letter sequence YNYYN. Since the number of 5-letter sequences such that each letter is either N or Y is 25, it follows that there are 25 = 32 subsets of {1, 2, 3, 4, 5}. For Statement (2), we are given that 2n = 16, and hence n = 4; SUFFICIENT.
Alternatively, observe that {1} has two subsets, {1, 2} has four subsets, and each addition of a new element doubles the number of subsets, because the subsets after adding the new element will consist of all the previous subsets along with the new element included in each of the previous subsets. Thus, {1, 2, 3} has 2(4) = 8 subsets, {1, 2, 3, 4} has 2(8) = 16 subsets, {1, 2, 3, 4, 5} has 2(16) = 32 subsets, etc.
The correct answer is D;each statement alone is sufficient.
Let n be the cardinality of the finite set A. What is the value of n ?
The number of 2-element subsets of A is equal to the number of unordered selections without replacement of 2 objects from a collection of n distinct objects, or “n choose 2.” Therefore, we have
= 6, or equivalently, n2 – n – 12 = 0. Because this is a quadratic equation that may have two solutions, we need to investigate further to determine whether there is a unique value of n. Factoring leads to (n – 4)(n + 3) = 0, and thus n = 4 or n = −3. Since n must be a nonnegative integer, we discard the solution n = −3. Therefore, n = 4; SUFFICIENT.
The number of subsets of set A is 2n, because each subset of A corresponds to a unique sequence of answers to yes-no questions about membership in the subset (one question for each of the n elements). For example, let A = {1, 2, 3, 4, 5}, let Y represent “yes,” and let N represent “no.” Then the sequence NYNNN corresponds to the subset {2}, since the answers to “is 1 in the subset,” “is 2 in the subset,” “is 3 in the subset,” etc. are “no,” “yes,” “no,” etc. Also, the subset {1, 3, 4} of A corresponds to the 5-letter sequence YNYYN. Since the number of 5-letter sequences such that each letter is either N or Y is 25, it follows that there are 25 = 32 subsets of {1, 2, 3, 4, 5}. For Statement (2), we are given that 2n = 16, and hence n = 4; SUFFICIENT.Alternatively, observe that {1} has two subsets, {1, 2} has four subsets, and each addition of a new element doubles the number of subsets, because the subsets after adding the new element will consist of all the previous subsets along with the new element included in each of the previous subsets. Thus, {1, 2, 3} has 2(4) = 8 subsets, {1, 2, 3, 4} has 2(8) = 16 subsets, {1, 2, 3, 4, 5} has 2(16) = 32 subsets, etc.
The correct answer is D;each statement alone is sufficient.
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