Question

Sketch the set \(X=[1,3] \times[1,2]\) on the plane \(\mathbb{R}^{2}\). On separate drawings, shade in the sets \(\bar{X}\) and \(\bar{X} \cap([0,2] \times[0,3])\)

Short answer

The set \(X=[1,3] \times[1,2]\) is a rectangle with corners at (1,1) and (3,2). Its compliment, \(\bar{X}\), is the entire plane, excluding the rectangle. The intersection \(\bar{X} \cap ([0,2] \times[0,3])\) is the area inside the rectangle defined by [0,2] \times [0,3] that does not overlap with X.

Step-by-step solution

Sketch the set \(X=[1,3] \times[1,2]\)

It's needed to sketch a coordinate system and then draw a rectangle with corners at the points (1,1) and (3,2). This rectangle is the set X.

Shade in the set \(\bar{X}\)

On a new sketch of the coordinate system, mark all the points that lie outside the rectangle from Step 1. The shaded region, which corresponds to the whole plane except the rectangle from Step 1, represents \(\bar{X}\).

Sketch and shade \(\bar{X} \cap ([0,2] \times[0,3])\)

Draw a new rectangle for the set \([0,2] \times[0,3]\) that corner points are (0,0) and (2,3). Then, shade the region inside this rectangle which does not overlap with X from step 1. This shaded region shows the intersection between \(\bar{X}\) and \([0,2] \times[0,3]\).