If \(A_i = \{-i, \dots , -2, -1, 0, 1, 2, \dots , i \}\) then…
2018
If \(A_i = \{-i, \dots , -2, -1, 0, 1, 2, \dots , i \}\)
then \(\cup_{i=1}^\infty A_i\) is :
- A.
Z
- B.
Q
- C.
R
- D.
C
Attempted by 497 students.
Show answer & explanation
Correct answer: A
Answer: The union is the set of all integers.
Definition: A_i = { -i, …, -1, 0, 1, …, i } contains only integers between -i and i.
Show union is contained in the integers: every A_i is a subset of the integers, so the union of all A_i is also a subset of the integers.
Show every integer is in the union: take any integer n. Let i = |n| (a positive integer). Then n is between -i and i, so n ∈ A_i. Hence n belongs to the union.
Conclusion: the union equals the set of all integers.
Example: 5 ∈ A_5, -3 ∈ A_3, while a number like 1/2 or √2 is not in any A_i.