- #1

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[tex]\mathbb{R}\times \mathbb{R}^2 = \mathbb{R}^2 \times \mathbb{R} = \mathbb{R}^3[/tex]

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In summary, the elements of the cartesian product of two sets, \mathbb{R}\times \mathbb{R}^2, are of the form (a, (b, c)), while the elements of the cartesian product of three sets, \mathbb{R}^2\times \mathbb{R}, are of the form ((a, b), c). However, they are considered isomorphic and are often referred to as simply \mathbb{R}^3 for convenience. This concept can also be expressed using the language of category theory.

- #1

- 4,807

- 32

[tex]\mathbb{R}\times \mathbb{R}^2 = \mathbb{R}^2 \times \mathbb{R} = \mathbb{R}^3[/tex]

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- #2

Hurkyl

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Yes. (with a but)

- #3

BicycleTree

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- #4

BicycleTree

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(Assuming you meant the 2 to have precedence over the x)

- #5

Hurkyl

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That's the essense of the "but".

- #6

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But how important is this "but"?

- #7

Hurkyl

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In the strictest sense, they aren't equal, but I don't think I've ever really seen anyone distinguish between them.

- #8

matt grime

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This is one of the "modern" ways of saying it in the lagauge of category theory.

A Cartesian Product is a mathematical operation that combines two sets to create a new set. It is denoted by the symbol x and is also known as a cross product. In terms of geometry, it represents the combination of two coordinate systems to create a new coordinate system.

The symbol "R^3" represents the set of all ordered triples of real numbers, which is also known as the three-dimensional Cartesian coordinate system. It is commonly used to represent points in three-dimensional space and is an essential concept in geometry, physics, and engineering.

The Cartesian Product is closely related to vectors as it can be used to represent and manipulate vectors in three-dimensional space. The cross product of two vectors results in a new vector that is perpendicular to both input vectors and has a magnitude equal to the product of their magnitudes.

The Cartesian Product has many practical applications in science, including physics, engineering, and computer graphics. It is used to represent the motion of objects in three-dimensional space, calculate forces and velocities, and create three-dimensional models of objects and structures.

Yes, the Cartesian Product can be extended to any number of dimensions. For example, the Cartesian Product of four sets would result in a four-dimensional space, and so on. This concept is used in higher mathematics and has applications in various fields of science and engineering.

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