Question

Sketch these Cartesian products on the \(x-y\) plane \(\mathbb{R}^{2}\) (or \(\mathbb{R}^{3}\) for the last two). $$ [0,1] \times[0,1] $$

Short answer

The Cartesian product [0,1] × [0,1] forms a unit square in the \(x-y\) plane with vertices at the points (0,0), (1,0), (0,1) and (1,1).

Step-by-step solution

Understanding the Cartesian Product

The Cartesian product of [0,1] and [0,1] is the set of all ordered pairs (x,y) such that \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\). These pairs represent points in the Cartesian plane.

Sketching the Cartesian Product

To sketch the Cartesian product of [0,1] and [0,1], we begin by drawing an \(x-y\) plane. Now, recall that for any (x, y), 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. Therefore, these pairs form a square in the plane with vertices at (0,0), (1,0), (0,1) and (1,1). Mark these points on the plane and then draw a line to connect them to form a square.