Question

Sketch the following sets of points in the \(x-y\) plane. $$ \\{(x, y): x \in[1,2], y \in[1,2]\\} $$

Short answer

The set forms a square in the Cartesian plane with vertices at points (1,1), (1,2), (2,1) and (2,2).

Step-by-step solution

Understand the Interval

The intervals given for \(x\) and \(y\) are both [1,2]. This means that \(x\) and \(y\) can take any value between 1 and 2, inclusive.

Plot Interval on the x-y plane

Start by plotting a point at the coordinates (1,1). Then, move horizontally (changing \(x\)) and move vertically (changing \(y\)) within the limits defined by the intervals. The shape that appears is a square which starts at the point (1,1) up to (2,2).

Draw the Square

The given intervals form a square in the Cartesian plane with vertices at points (1,1), (1,2), (2,1) and (2,2). Draw lines connecting these points to complete the square.