If R is the set of all real numbers, what do the cartesian products R×R and R×R×R represent?
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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 R×R
For the Cartesian product R×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 R×R×R
Similarly, for R×R×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.