Question

(a) \(\bigcup_{\alpha \in \mathbb{R}}\\{\alpha\\} \times[0,1]=\) (b) \(\bigcap_{\alpha \in \mathbb{R}}\\{\alpha\\} \times[0,1]=\)

Short answer

(a) \(\mathbb{R} \times [0,1]\) \n(b) \( \emptyset \)

Step-by-step solution

Understand the Cartesian Product

A Cartesian product is created by pairing elements of two sets. Every pair includes one element from each set. So, the Cartesian product of the set \({\alpha}\) and the interval [0,1] is a set of ordered pairs: each pair starts with the number \(\alpha\), and its second element is a number from the interval [0,1].

Comprehend the Union Over All \(\mathbb{R}\)

The union operation joins together all the sets. In part (a), we are asked to find the union of these Cartesian products for all \(\alpha\) in \(\mathbb{R}\). That would mean we would gather all possible pairs where the first element is any real number and the second element is a number in [0,1]. Because we allow any real number for \(\alpha\), the result is the set of all pairs (\(x,y\)) where \(x\) is any real number and \(y\) is a number in [0,1].

Evaluate the Intersection Over All \(\mathbb{R}\)

The intersection operation finds common elements among all sets. In part (b), unlike the union operation, the intersection between all the Cartesian products would result in an empty set, that is because there is no ordered pair that would exist in all sets, as each pair in the Cartesian product has an unique \(\alpha\) as a first element.