Question

Is the statement \((\mathbb{R} \times \mathbb{Z}) \cap(\mathbb{Z} \times \mathbb{R})=\mathbb{Z} \times \mathbb{Z}\) true or false? What about the statement \((\mathbb{R} \times \mathbb{Z}) \cup(\mathbb{Z} \times \mathbb{R})=\mathbb{R} \times \mathbb{R} ?\)

Short answer

The first statement \((\mathbb{R} \times \mathbb{Z}) \cap(\mathbb{Z} \times \mathbb{R})=\mathbb{Z} \times \mathbb{Z}\) is true, while the second statement \((\mathbb{R} \times \mathbb{Z}) \cup(\mathbb{Z} \times \mathbb{R})=\mathbb{R} \times \mathbb{R}\) is false.

Step-by-step solution

Understanding the Terms

Firstly, analyze and get a clear understanding about the Cartesian product, union, and intersection. Cartesian product of two sets A and B, denoted by A x B, is the set of all ordered pairs (a, b) where a is in A and b is in B. Intersection of two sets A and B, denoted A ∩ B, is a set containing all the common elements of A and B. Union of two sets A and B, denoted A ∪ B, is a set containing all the elements of A and B.

Evaluate the First Statement

The statement \((\mathbb{R} \times \mathbb{Z}) \cap(\mathbb{Z} \times \mathbb{R})=\mathbb{Z} \times \mathbb{Z}\) refers to the intersection of two Cartesian products, which yields the set of all common ordered pairs. Here, the first Cartesian product contains ordered pairs where the first element is any real number and the second element is an integer. Similarly, the second Cartesian product includes the pairs where the first element is an integer and the second element is a real number. \(\mathbb{Z} \times \mathbb{Z}\) is the set of ordered pairs of integers. So, for the intersection, both elements in the ordered pair must be integers. This means the statement is correct.

Evaluate the Second Statement

The statement \((\mathbb{R} \times \mathbb{Z}) \cup(\mathbb{Z} \times \mathbb{R})=\mathbb{R} \times \mathbb{R}\) refers to the union of two Cartesian products. The union will include all elements from both sets. Here, neither Cartesian product covers all possible pairs of real numbers because one element in each pair is always an integer. Thus, their union can't form \(\mathbb{R} \times \mathbb{R}\), the set of all ordered pairs of real numbers. This means the statement is false.