User blog:Holomanga/Sets and Dimensions

What is a set?
A set is a collection of mathematical objects. Usually, these are numbers, but they can also be shapes, functions, and even other sets. $$\{ 5,7,3\} $$ is an example of a set.

Each object inside a set is called an element of the set. For example, 3 is an element of the example set above.

The total number of elements in the set is the cardinality of the set. The cardinality of the example set above is 3, since it has 3 elements.

Empty Set
$$\mathbb{ \emptyset }$$ The empty set is the set containing no elements, represented by the symbol $$\mathbb{ \emptyset }$$. It has a cardinality of 0, since it contains no elements.

The empty set has a few applications in higher-dimensional shapes. With abstract polytopes, it corresponds to the null polytope, and is -1 dimensional.

Boolean Set
$$\mathbb{ B }$$ The boolean set, represented by the symbol $$\mathbb{ B }$$, is the set $$\{0, 1\}$$. It has 2 elements, and hence has a cardinality of 2.

Power Set of Booleans
$${2}^\mathbb{ B }$$

About now is time to introduce you to the idea of power sets. The power set of a set is the set of all it's subsets. The power set of set X is written as $${2}^{X}$$. With the boolean set, $${2}^{\mathbb{ B }} = \{ \{ 0,1\} ,\{ 0\} ,\{ 1\} ,\emptyset \} $$.

You may notice that the cardinality of $${2}^\mathbb{ B }$$ is $${2}^{2}$$ (4). This stands as a general rule; $$\left| { 2 }^{ X } \right| ={ 2 }^{ \left| X \right| }$$ for any set X.

Set of Functions mapping Booleans to Booleans
$$\{ f|\mathbb{ B }\xrightarrow { f } \mathbb{ B }\}$$

Also to the idea of functions, as they apply to sets. A function defined over $$X\xrightarrow { f } Y$$ maps elements of set X to elements of set Y. The cardinality of the set of all functions $$X\xrightarrow { f } Y$$ is equal to $${ \left| Y \right| }^{ \left| X \right|  }$$

This means that there are four functions which map a single value in $$\mathbb{ B }$$ to another single value in $$\mathbb{ B }$$, since $${ \left| { \mathbb{ B }  } \right|  }^{ \left|  \mathbb{ B }  \right|  }={ 2 }^{ 2 }=4$$. These four functions can be easily listed (here, $${b}_{n}$$ denotes an element of $$\mathbb{ B }$$): Note that these correspond to the elements of $${2}^\mathbb{ B }$$ These are the only functions allowed; any other function that you could possibly think of has some result that isn't an element of $$\mathbb{ B }$$.
 * 1) Tautology Function: $$f(b)=1$$
 * 2) Identity Function: $$f(b)=b$$
 * 3) Negation Function: $$f(b)=1-b=\neg b$$
 * 4) Contradiction Function: $$f(b)=0$$

Set of Two Booleans
$${\mathbb{ B }}^{2}$$ The boolean squared set, also known as the set of two booleans, represented by the symbol $${\mathbb{ B }}^{2}$$, represents the set of pairs of elements of $$\mathbb{ B }$$. Written out fully, it is the set $$\{ (0,0),(0,1),(1,0),(1,1)\}$$. This set has a cardinality of 4, which you might notice as equal to $${\left| \mathbb{ B } \right| }^{2}$$. This also applies generally.

Set of Functions Mapping Two Booleans to Booleans
$$\{ f|{\mathbb{ B }}^{2}\xrightarrow { f } \mathbb{ B }\}$$

Coming Soon