> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/200-set-theory/topics/set-types.md).

# Set Types

Sets:

* *pure set*: a set containing scalars only
* *empty set*: pretty vacant
* *higher-order set*: a set containing set(s)
* *mixed set*:a set known to contain at least one scalar AND at least one set
* *universal set*: set containing everything, the universe

Types:

* Ordered pair or pair
* Powerset
* Cartesian product or cross product
* Partitioning
* Bell Numbers

wrt cardinality:

* 0: Empty set
* 1: Singleton set or unit set
* 2: Unordered pair
* Finite set
* Infinite set
* Universal set

wrt relations:

* Equal sets
* Equivalent sets
* Overlapping sets
* Disjoint sets

wrt set operations:

* Union
* Intersection
* Difference
* Relative complement

Properties:

* Commutative
* Associative
* Distributive
* Idempotency
* Identity
* Transitive
* Involution
* De Morgan's Law
* Closure

## Types of Sets

Sets can be classified into many types, including: finite, infinite, universal, singleton, empty set.

**Proper Subset** A Set X is a proper subset of set Y (Written as X⊂Y) if every element of X is an element of set Y and |X|<|Y|.

X={1,2,3,4,5,6} and Y={1,2}. Here set Y⊂X since all elements in Y are contained in X too and X has at least one element is more than set Y.

**Universal Set** It is a collection of all elements in a particular context or application. All the sets in that context or application are essentially subsets of this universal set. Universal sets are represented as U.

Example − We may define U as the set of all animals on earth. In this case, set of all mammals is a subset of U, set of all fishes is a subset of U, set of all insects is a subset of U, and so on.

**Singleton Set** or **Unit Set** Singleton set or unit set contains only one element. A singleton set is denoted by {s}. Example − S={x|x∈N, 7\<x<9} = {8}

**Equal Set** If two sets contain the same elements they are said to be equal. Example − If A={1,2,6} and B={6,1,2}, they are equal as every element of set A is an element of set B and every element of set B is an element of set A.

**Equivalent Set** If the cardinalities of two sets are same, they are called equivalent sets. Example − If A={1,2,6} and B={16,17,22}, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A|=|B|=3

**Overlapping Set** Two sets that have at least one common element are called overlapping sets. In case of overlapping sets − n(A∪B)=n(A)+n(B)−n(A∩B) n(A∪B)=n(A−B)+n(B−A)+n(A∩B) n(A)=n(A−B)+n(A∩B) n(B)=n(B−A)+n(A∩B) Example − Let, A={1,2,6} and B={6,12,42}. There is a common element ‘6’, hence these sets are overlapping sets.

**Disjoint Set** Two sets A and B are called disjoint sets if they do not have even one element in common. Therefore, disjoint sets have the following properties − n(A∩B)=∅ n(A∪B)=n(A)+n(B) Example − Let, A={1,2,6} and B={7,9,14}, there is not a single common element, hence these sets are overlapping sets.

**Power set** of a set S is the set of all subsets of S including the empty set. The cardinality of a power set of a set S of cardinality n is 2^n

## Set Operations

* Union
* Intersection
* Difference
* Complement
* Cartesian (cross) product

## Partitioning

Partition of a set, say S, is a collection of n disjoint subsets, say P1,P2,…Pn that satisfies the following three conditions −

Pi does not contain the empty set.

\[Pi≠{∅} for all 0\<i≤n] The union of the subsets must equal the entire original set.

\[P1∪P2∪⋯∪Pn=S] The intersection of any two distinct sets is empty.

\[Pa∩Pb={∅}, for a≠b where n≥a,b≥0] Example

Let S={a,b,c,d,e,f,g,h} One probable partitioning is {a},{b,c,d},{e,f,g,h} Another probable partitioning is {a,b},{c,d},{e,f,g,h}

<https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematics_quick_guide.htm>