Distribution

This article explains the distribution of numbers, a mathematical concept that is required by the pilitron theory.

A distribution is simply a set of numbers or other values that operations can be performed on, e.g. vectors or scalars. This set my be finite or infinite. What is important is not the idea that a distribution is simply a set of numbers, but the rules that apply to distrbutions. We will start with a simple distribution, $$\mathsf{X}$$:


 * $$\mathsf{X} = \left[1, 2, 3\right]$$

When I perform an operation on the distrbution, I get another distribution, whose contents are the same as the original distributions, but with the operation performed on them. Example:


 * $$\mathsf{X}^2 = \left[1^2, 2^2, 3^2\right] = \left[1, 4, 9\right]$$

The order of components inside of a distribution is irrelevant.

Exclusions
Components must be excluded from a distribution in certain circumstances.

Operational Exclusion
One rule is that if an operation cannot be performed on some component of the distribution, the resulting distributions lacks this component. For example, let's take a distribution $$\mathsf{X}$$:


 * $$\mathsf{X} = \left[-1, 0, 1, 2\right]$$

And now when we try to find the reciprocal of it, we get:


 * $$\frac{1}{\mathsf{X}} = \left[\frac{1}{-1}, \frac{1}{0}, \frac{1}{1}, \frac{1}{2}\right]$$

As you can see, we have to calculate $$1/0$$, which is impossible. Therefore, we simply exclude this component from the distribution and get:


 * $$\frac{1}{\mathsf{X}} = \left[-1, 1, \frac{1}{2}\right]$$

Competitive Exclusion
Each component can only appear once inside of a distribution. Therefore, if there are five '2's, only one of them stays.

Contradictional Exclusion
This is a type of explicit exclusion.

It is denoted as follows:


 * $$\mathsf{X} = \text{excl}\left(x \leftarrow \mathsf{S}\right) \left[x > 5\right] \text{AND} \left[x < 16 \right]$$

This means that the distribution $$\mathsf{X}$$ contains all components of $$\mathsf{S}$$ but only if the given component $$x$$ is larger than 5 and smaller than 16. The operator $$\text{OR}$$ (with obvious meaning) can also be used.

Inclusion
Inclusion is also an explicit exclusion.

It is denoted by $$\mathsf{A} \times \mathsf{B}$$ and simply returns a distribution that contains everything that is both in $$\mathsf{A}$$ and in $$\mathsf{B}$$, and nothing else.

Special distributions
The following distributions are considered special:


 * $$\ddot{\mathsf{R}}$$ contains all real numbers.
 * $$\ddot{\mathsf{C}}$$ contains all complex numbers.
 * $$\ddot{\mathsf{I}}$$ contains all integers.
 * $$\ddot{\mathsf{U}}$$ is the unit distribution, $$\left[0, 1\right]$$, the answer to $$0^0$$.