If $\mathbf{R}$ is the set of all real numbers, what do the cartesian products $\mathbf{R} \times \mathbf{R}$ and $\mathbf{R} \times \mathbf{R} \times \mathbf{R}$ represent?

Answer: The Cartesian product $\mathbf{R} \times \mathbf{R}$ represents the two-dimensional Cartesian coordinate plane, and $\mathbf{R} \times \mathbf{R} \times \mathbf{R}$ represents the three-dimensional Cartesian coordinate space.

Step-by-step solution

Step 1: Understand Cartesian Product

The Cartesian product of two sets, say $A$ and $B$ , is the set of all possible ordered pairs $(a, b)$ where the first element $a$ comes from set $A$ and the second element $b$ comes from set $B$ . This concept extends to more than two sets.

Step 2: Interpret $\mathbf{R} \times \mathbf{R}$

For the Cartesian product $\mathbf{R} \times \mathbf{R}$ , both sets are the set of all real numbers. This means each ordered pair $(x, y)$ consists of two real numbers. Geometrically, these ordered pairs represent all points in a two-dimensional Cartesian coordinate plane.

Step 3: Interpret $\mathbf{R} \times \mathbf{R} \times \mathbf{R}$

Similarly, for $\mathbf{R} \times \mathbf{R} \times \mathbf{R}$ , the elements are ordered triplets $(x, y, z)$ , where each component is a real number. These ordered triplets represent all points in a three-dimensional Cartesian coordinate space.

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If R\mathbf{R}R is the set of all real numbers, what do the cartesian products R×R\mathbf{R} \times \mathbf{R}R×R and R×R×R\mathbf{R} \times \mathbf{R} \times \mathbf{R}R×R×R represent?