让整个空间都是小于 10 的正整数。设 A={1, 2, 3, 4, 5}，Ac（A 的补集是）？

Let the entire space be all positive integers smaller than 10. Given A={1, 2, 3, 4, 5}, what is Ac (the complement of A)?

Question

让整个空间都是小于 10 的正整数。设 A={1, 2, 3, 4, 5}，Ac（A 的补集是）？

Let the entire space be all positive integers smaller than 10. Given A={1, 2, 3, 4, 5}, what is Ac (the complement of A)?

BlackTom AI · Accepted Answer

The complement consists of all elements in the universal set that are not in A. Here the universal set is the positive integers less than 10, so U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.

1. Relevant concepts
- Universal set U: all elements under consideration (given as positive integers < 10).
- Subset A: A = {1, 2, 3, 4, 5}.
- Complement A^c (or Ac): A^c = U \ A = all elements of U that are not in A.

2. Step-by-step reasoning
- Step 1: Write the universal set explicitly: U = {1,2,3,4,5,6,7,8,9}.
- Step 2: List A: A = {1,2,3,4,5}.
- Step 3: Remove from U every element that appears in A. Those removed are 1, 2, 3, 4, 5.
- Step 4: The remaining elements are 6, 7, 8, 9.

3. Intermediate steps (set subtraction)
U \ A = {1,2,3,4,5,6,7,8,9} \ {1,2,3,4,5}
= {6,7,8,9}.

4. Why this is correct
Each element of {6,7,8,9} is in the universal set and not in A, so they belong to A^c. Every element of A (1–5) is excluded by definition of complement. No other positive integers less than 10 remain.

5. Why other potential options would be incorrect
- Any option that includes numbers 1–5 would be wrong because those are in A, not in A^c.
- Any option including 0 or 10 (or larger) would be wrong because they are not in the universal set (positive integers < 10).
- The empty set is wrong because there are elements of U not in A.

Conclusion: The complement A^c = {6, 7, 8, 9}, which matches the provided answer.

让整个空间都是小于 10 的正整数。设 A={1, 2, 3, 4, 5}，Ac（A 的补集是）？ Let the entire space be all positive integers smaller than 10. Given A={1, 2, 3, 4, 5}, what is Ac (the complement of A)? 单项选择题

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