## 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. 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

The empty set is the set containing no elements, represented by the symbol . 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

The boolean set, represented by the symbol , is the set . It has 2 elements, and hence has a cardinality of 2.

### Power Set of Booleans

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 . With the boolean set, .

You may notice that the cardinality of is (4). This stands as a general rule; for any set X.

### Set of Functions mapping Booleans to Booleans

Also to the idea of functions, as they apply to sets. A **function** defined over maps elements of set X to elements of set Y.
The cardinality of the set of all functions is equal to

This means that there are four functions which map a single value in to another single value in , since . These four functions can be easily listed (here, denotes an element of ):

- Tautology Function:
- Identity Function:
- Negation Function:
- Contradiction Function:

Note that these correspond to the elements of These are the only functions allowed; any other function that you could possibly think of has some result that isn't an element of .

### Set of Two Booleans

The boolean squared set, also known as the set of two booleans, represented by the symbol , represents the set of pairs of elements of . Written out fully, it is the set . This set has a cardinality of 4, which you might notice as equal to . This also applies generally.

### Set of Functions Mapping Two Booleans to Booleans

*Coming Soon*