Question

For each \(n \in \mathbb{N},\) let \(A_{n}=\\{0,1,2,3, \ldots, n\\}\) (a) \(\bigcup_{i \in \mathbb{N}} A_{i}=\) (b) \(\bigcap_{i \in N} A_{i}=\)

Short answer

(a) \(\mathbb{N}\), (b) \(\{0\}\)

Step-by-step solution

Understanding the Sets

Note that each \(A_{n}\) consists of the set of natural numbers from 0 to \(n\). So, \(A_{1}\) is \( \{0, 1\}\), \(A_{2}\) is \( \{0, 1, 2\}\), and so forth.

Calculate the Union

The union of all sets \(A_{n}\) is the set of all elements that appear in any \(A_{n}\) set. Since every natural number appears in some \(A_{n}\) (namely, in \(A_{n}\) where \(n\) is the number itself), the union result is the set of all natural numbers, that is, \(\mathbb{N}\).

Calculate the Intersection

The intersection of all sets \(A_{n}\) is the set of all elements that appear in all sets \(A_{n}\). Here, note that \(A_{n}\) only contains numbers up to \(n\). Thus, as \(n\) increases, only 0 is commonly found in all the sets. Hence, the intersection is the set containing the single element 0, denoted by \(\{0\}\).