Question

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

Short answer

The union of the sets (a) is the set \(R \times [1, +\infty)\) and the intersection of the sets (b) is an empty set \(\emptyset\).

Step-by-step solution

Understanding the symbols

The symbol \(\bigcup\) represents the union of sets and \(\bigcap\) represents the intersection of sets. \(R\) represents the set of all real numbers, and \(i\) represents an element of natural numbers. \([i, i+1]\) is a continuous closed interval from \(i\) to \(i+1\). \(R \times[i, i+1]\) is the Cartesian product between the set of all real numbers and the closed interval \([i, i+1]\).

Part (a) - Calculate the Union

The union over all \(i\) in \(\mathbb{N}\) of the set \(R \times[i, i+1]\) forms the set of all possible ordered pairs (real number, real number in the closed interval) for all \(i\). This can be visualized as an infinitely large set of vertical strips in the two-dimensional plane. Since the intervals \([i, i+1]\) cover the entire positive real number line for all \(i\) in \(\mathbb{N}\), the union over all \(i\) in \(\mathbb{N}\) will cover the set of all real numbers and all positive real numbers. So, \(\bigcup_{i \in \mathbb{N}} R \times[i, i+1] = R \times [1, +\infty)\)

Part (b) - Calculate the Intersection

The intersection over all \(i\) in \(\mathbb{N}\) of the set \(R \times[i, i+1]\) forms the set of ordered pairs where every pair is shared among all the sets for all \(i\). Since each set \(R \times[i, i+1]\) contains different continuous intervals, when considering the intersection over all \(i\), there will be no common elements. In other words, no order pair belongs to all these sets simultaneously. Hence, \(\bigcap_{i \in \mathbb{N}} R \times[i, i+1] = \emptyset\) (the empty set).