Question

Sketch these Cartesian products on the \(x-y\) plane \(\mathbb{R}^{2}\) (or \(\mathbb{R}^{3}\) for the last two). $$ \\{1,1.5,2\\} \times[1,2] $$

Short answer

The Cartesian product \{1,1.5,2\} \times [1,2] when plotted on the \(x-y\) plane will result in three vertical line segments. Each line originates from the x-values 1, 1.5, and 2 respectively and range from y-values 1 to 2.

Step-by-step solution

Identify the elements of the Cartesian product

The Cartesian product of the given sets will include every combination of one element from the first set and one element from the second set. Because the second set is a continuous interval, each combination will include one number from the first set and all real numbers between 1 and 2 from the second set. So, the Cartesian product can be written as \{(1,y)| 1\leq y \leq 2\}, \{(1.5,y)| 1\leq y \leq 2\}, and \{(2,y)| 1\leq y \leq 2\}.

Plot the Cartesian product on a \(x-y\) plane

On the \(x-y\) plane, for every corresponding x-value (from the first set: 1, 1.5, 2), draw a vertical line segment between y = 1 and y = 2. These lines represent all the ordered pairs in the Cartesian product. Each point on a line corresponds to an ordered pair in the Cartesian product.