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

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