Question

(a) \(\bigcup_{i \in \mathbb{N}}[i, i+1]=\) (b) \(\bigcap_{i \in \mathbb{N}}[i, i+1]=\)

Short answer

The union of the intervals is [1, +∞) and the intersection of the intervals is ∅.

Step-by-step solution

Union of Sets

To find the union (denoted as ∪), locate the area on the number line that is covered by any one of the sets. The union of the given intervals [i, i+1] for i∈ℕ{s where ℕ represents the set of all natural numbers includes all points from each integer i and onward. This is because for each i in ℕ, the interval [i, i+1] covers the range of that integer and one unit onward. So, the union over all i will cover the full set of natural numbers onwards. Hence, the union (denoted by '∪') of all intervals [i, i+1] for i∈ℕ is represented by [1, +∞).

Intersection of Sets

To find the intersection (denoted by '∩'), locate the area on the number line that is covered by all of the sets. However, in this case, there is no single real number that belongs to all the intervals [i, i+1] for i in ℕ simultaneously. This is because the start of each interval increases as i increase. Consequently, there is no common region in all intervals. Hence, the intersection of all the intervals [i, i+1] for i∈ℕ is represented by an empty set ∅.